Element structure of groups of order 32

This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 32.
View element structure of group families | View element structure of groups of a particular order |View other specific information about groups of order 32

Conjugacy class sizes

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

Full listing

Here is the conjugacy class structure for all the groups of order 32:

[SHOW MORE]

Group	Second part of GAP ID (GAP ID is (32,second part))	Hall-Senior number (among groups of order 32)	Hall-Senior symbol	Nilpotency class	Number of conjugacy classes of size 1	Number of conjugacy classes of size 2	Number of conjugacy classes of size 4	Number of conjugacy classes of size 8
Cyclic group:Z32	1	7	$(5)$	1	32	0	0	0
SmallGroup(32,2)	2	18	$\Gamma_2h$	2	8	12	0	0
Direct product of Z8 and Z4	3	5	$(32)$	1	32	0	0	0
Semidirect product of Z8 and Z4 of M-type	4	19	$\Gamma_2i$	2	8	12	0	0
SmallGroup(32,5)	5	20	$\Gamma_2j$	2	8	12	0	0
Faithful semidirect product of E8 and Z4	6	46	$\Gamma_7a_1$	3	2	3	6	0
SmallGroup(32,7)	7	47	$\Gamma_7a_2$	3	2	3	6	0
SmallGroup(32,8)	8	48	$\Gamma_7a_3$	3	2	3	6	0
SmallGroup(32,9)	9	27	$\Gamma_3c_1$	3	4	6	4	0
SmallGroup(32,10)	10	28	$\Gamma_3c_2$	3	4	6	4	0
Wreath product of Z4 and Z2	11	31	$\Gamma_3e$	3	4	6	4	0
SmallGroup(32,12)	12	21	$\Gamma_2j_2$	2	8	12	0	0
Semidirect product of Z8 and Z4 of semidihedral type	13	30	$\Gamma_3d_2$	3	4	6	4	0
Semidirect product of Z8 and Z4 of dihedral type	14	29	$\Gamma_3d_1$	3	4	6	4	0
SmallGroup(32,15)	15	32	$\Gamma_3e$	3	4	6	4	0
Direct product of Z16 and Z2	16	6		1	32	0	0	0
M32	17	22	$\Gamma_2k$	2	8	12	0	0
Dihedral group:D32	18	49	$\Gamma_8a_1$	4	2	7	0	2
Semidihedral group:SD32	19	50	$\Gamma_8a_2$	4	2	7	0	2
Generalized quaternion group:Q32	20	51	$\Gamma_8a_3$	4	2	7	0	2
Direct product of Z4 and Z4 and Z2	21	3	$(2^21)$	1	32	0	0	0
Direct product of SmallGroup(16,3) and Z2	22	11	$\Gamma_2c_1$	2	8	12	0	0
Direct product of SmallGroup(16,4) and Z2	23	12	$\Gamma_2c_2$	2	8	12	0	0
SmallGroup(32,24)	24	16	$\Gamma_2f$	2	8	12	0	0
Direct product of D8 and Z4	25	14	$\Gamma_2e_1$	2	8	12	0	0
Direct product of Q8 and Z4	26	15	$\Gamma_2e_2$	2	8	12	0	0
SmallGroup(32,27)	27	33	$\Gamma_4a_1$	2	4	6	4	0
SmallGroup(32,28)	28	36	$\Gamma_4b_1$	2	4	6	4	0
SmallGroup(32,29)	29	37	$\Gamma_4b_2$	2	4	6	4	0
SmallGroup(32,30)	30	38	$\Gamma_4c_1$	2	4	6	4	0
SmallGroup(32,31)	31	39	$\Gamma_4c_2$	2	4	6	4	0
SmallGroup(32,32)	32	40	$\Gamma_4c_3$	2	4	6	4	0
SmallGroup(32,33)	33	41	$\Gamma_4d$	2	4	6	4	0
Generalized dihedral group for direct product of Z4 and Z4	34	34	$\Gamma_4a_2$	2	4	6	4	0
SmallGroup(32,35)	35	35	$\Gamma_4a_3$	2	4	6	4	0
Direct product of Z8 and V4	36	4		1	32	0	0	0
Direct product of M16 and Z2	37	13	$\Gamma_2d$	2	8	12	0	0
SmallGroup(32,38)	38	17	$\Gamma_2g$	2	8	12	0	0
Direct product of D16 and Z2	39	23	$\Gamma_3a_1$	3	4	6	4	0
Direct product of SD16 and Z2	40	24	$\Gamma_3a_2$	3	4	6	4	0
Direct product of Q16 and Z2	41	25	$\Gamma_3a_3$	3	4	6	4	0
Central product of D16 and Z4	42	26	$\Gamma_3b$	3	4	6	4	0
Holomorph of Z8	43	44	$\Gamma_6a_1$	3	2	3	6	0
SmallGroup(32,44)	44	45	$\Gamma_6a_2$	3	2	3	6	0
Direct product of E8 and Z4	45	2	$(21^3)$	1	32	0	0	0
Direct product of D8 and V4	46	8	$\Gamma_2a_1$	2	8	12	0	0
Direct product of Q8 and V4	47	9	$\Gamma_2a_2$	2	8	12	0	0
Direct product of SmallGroup(16,13) and Z2	48	10	$\Gamma_2b$	2	8	12	0	0
Inner holomorph of D8	49	42	$\Gamma_5a_1$	2	2	15	0	0
SmallGroup(32,50)	50	43	$\Gamma_5a_2$	2	2	15	0	0
Elementary abelian group:E32	51	1	$(1^5)$	1	32	0	0	0

Grouping by conjugacy class sizes

Here now is a grouping by conjugacy class sizes. Note that since number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one, all the values for the number of conjugacy classes are congruent to 32 mod 3, and hence congruent to 2 mod 3.

Number of conjugacy classes of size 1	Number of conjugacy classes of size 2	Number of conjugacy classes of size 4	Number of conjugacy classes of size 8	Total number of conjugacy classes	Total number of groups	Nilpotency class(es) attained by these	Description of groups	List of groups	List of GAP IDs (ascending order)	List of Hall-Senior numbers (ascending order)	List of Hall-Senior symbols/families
32	0	0	0	32	7	1	all the abelian groups of order 32	cyclic group:Z32, direct product of Z8 and Z4, direct product of Z16 and Z2, direct product of Z4 and Z4 and Z2, direct product of Z8 and V4, direct product of E8 and Z4, elementary abelian group:E32	1, 3, 16, 21, 36, 45, 51	1--7	$\Gamma_1$ (abelian)
8	12	0	0	20	15	2	inner automorphism group is Klein four-group and derived subgroup is cyclic group:Z2	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	2, 4, 5, 12, 17, 22, 23, 24, 25, 26, 37, 38, 46, 47, 48	8--22	$\Gamma_2$
4	6	4	0	14	19	2,3	?	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	9, 10, 11, 13, 14, 15, 27, 28, 29, 30, 31, 32, 33, 34, 35, 39, 40, 41, 42	23--41	$\Gamma_3$ (class three, ten groups numbers 23--32) and $\Gamma_4$ (class two, nine groups, numbers 33--41)
2	15	0	0	17	2	2	the two extraspecial groups	inner holomorph of D8, central product of D8 and Q8	49, 50	42, 43	$\Gamma_5$ (the Hall-Senior family for extraspecial groups)
2	3	6	0	11	5	3	?	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	6, 7, 8, 43, 44	44--48	$\Gamma_6$ and $\Gamma_7$
2	7	0	2	11	3	4	the maximal class groups	dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32	18, 19, 20	49--51	$\Gamma_8$

Grouping by cumulative conjugacy class sizes (number of elements)

Number of elements in (size dividing 1) conjugacy classes	Number of elements in (size dividing 2) conjugacy classes	Number of elements in (size dividing 4) conjugacy classes	Number of elements in (size dividing 8) conjugacy classes	Total number of conjugacy classes	Total number of groups	Nilpotency class(es) attained by these groups	Description of groups	List of groups	List of GAP IDs (ascending order)	List of Hall-Senior numbers (ascending order)	List of Hall-Senior symbols/families
32	32	32	32	32	7	1	all the abelian groups of order 32	cyclic group:Z32, direct product of Z8 and Z4, direct product of Z16 and Z2, direct product of Z4 and Z4 and Z2, direct product of Z8 and V4, direct product of E8 and Z4, elementary abelian group:E32	1, 3, 16, 21, 36, 45, 51	1--7	$\Gamma_1$ (abelian)
8	32	32	32	20	15	2	inner automorphism group is Klein four-group and derived subgroup is cyclic group:Z2	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	2, 4, 5, 12, 17, 22, 23, 24, 25, 26, 37, 38, 46, 47, 48	8--22	$\Gamma_2$
4	16	32	32	14	19	2,3	?	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	9, 10, 11, 13, 14, 15, 27, 28, 29, 30, 31, 32, 33, 34, 35, 39, 40, 41, 42	23--41	$\Gamma_3$ (class three, ten groups numbers 23--32) and $\Gamma_4$ (class two, nine groups, numbers 33--41)
2	32	32	32	17	2	2	the two extraspecial groups	inner holomorph of D8, central product of D8 and Q8	49, 50	42, 43	$\Gamma_5$ (the Hall-Senior family for extraspecial groups)
2	8	32	32	11	5	3	?	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	6, 7, 8, 43, 44	44--48	$\Gamma_6$ and $\Gamma_7$
2	16	16	32	11	3	4	the maximal class groups	dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32	18, 19, 20	49--51	$\Gamma_8$

It is true for this order that the cumulative conjugacy class size statistics values divide the order of the group in all cases. In fact, this is true in general when the order is $p^k$, with $p$ a prime and $0 \le k \le 5$. There are, however, counterexamples for $2^6$.

• There exist groups of prime-sixth order in which the cumulative conjugacy class size statistics values do not divide the order of the group
• All cumulative conjugacy class size statistics values divide the order of the group for groups up to prime-fifth order

Correspondence between conjugacy class sizes and degrees of irreducible representations

See also linear representation theory of groups of order 32#Degrees of irreducible representations

For groups of order 32, it is true that the list of conjugacy class sizes completely determines the list of degrees of irreducible representations, and vice versa. The details are given below. The middle column, which is the total number of each, separates the description of the list of conjugacy class sizes and the list of degrees of irreducible representations:

Number of conjugacy classes of size 1	Number of conjugacy classes of size 2	Number of conjugacy classes of size 4	Number of conjugacy classes of size 8	Total number of conjugacy classes	Number of irreps of degree 1	Number of irreps of degree 2	Number of irreps of degree 4
32	0	0	0	32	32	0	0
8	12	0	0	20	16	4	0
2	15	0	0	17	16	0	1
4	6	4	0	14	8	6	0
2	3	6	0	11	8	2	1
2	7	0	2	11	4	7	0

Facts illustrated by these listings

• Nilpotency class and order need not determine conjugacy class size statistics for groups of prime-fifth order
• Number of conjugacy classes need not determine conjugacy class size statistics for groups of prime-fifth order
• Conjugacy class size statistics need not determine nilpotency class for groups of prime-fifth order

1-isomorphism

Pairs where one of the groups is abelian

There are eight pairs of groups that are 1-isomorphic with the property that one of them is abelian. Of these, some pairs share the abelian group part.

Here is a summary version:

Nature of 1-isomorphism	Intermediate object	Number of 1-isomorphisms between non-abelian and abelian group of this type	Number of 1-isomorphisms between non-abelian and abelian group of this nature, not of any of the preceding types	Note
linear halving generalization of Baer correspondence	class two Lie ring	0	0	The smallest examples for that are among groups of order 64. See element structure of groups of order 64#1-isomorphism.
cocycle halving generalization of Baer correspondence	class two Lie cring	6	6	See table below.
cocycle skew reversal generalization of Baer correspondence	class two near-Lie cring	7	1	The new example is SmallGroup(32,2). See table below.
linear halving generalization of Lazard correspondence	class three Lie ring	1	0	The unique example for this, namely semidirect product of Z8 and Z4 of M-type (ID: (32,4)), is alternatively explained using the cocycle halving generalization of Baer correspondence, so no new examples are added here.
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	8	1	This is SmallGroup(32,33).

Here are the details:

Non-abelian member of pair	GAP ID	Hall-Senior symbol	Hall-Senior number	Abelian member of pair	GAP ID	The 1-isomorphism arises as a ...	Description of the 1-isomorphism	Best perspective 1	Best perspective 2	Alternative perspective
M32	17	$\Gamma_2k$	22	direct product of Z16 and Z2	16	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring	generalized Baer correspondence between M32 and direct product of Z16 and Z2	second cohomology group for trivial group action of V4 on Z8#Generalized Baer Lie rings	second cohomology group for trivial group action of direct product of Z4 and Z2 on Z4#Generalized Baer Lie rings	second cohomology group for trivial group action of direct product of Z4 and Z4 on Z2
semidirect product of Z8 and Z4 of M-type	4	$\Gamma_2i$	19	direct product of Z8 and Z4	3	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring; alternatively, the linear halving generalization of Lazard correspondence, the intermediate object being a class three Lie ring	generalized Baer correspondence between semidirect product of Z8 and Z4 of M-type and direct product of Z8 and Z4	second cohomology group for trivial group action of V4 on direct product of Z4 and Z2#Generalized Baer Lie rings	second cohomology group for trivial group action of direct product of Z4 and Z2 on Z4#Generalized Baer Lie rings	second cohomology group for trivial group action of direct product of Z4 and Z4 on Z2
direct product of M16 and Z2	37	$\Gamma_2d$	13	direct product of Z8 and V4	36	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring	generalized Baer correspondence between direct product of M16 and Z2 and direct product of Z8 and V4	second cohomology group for trivial group action of V4 on direct product of Z4 and Z2#Generalized Baer Lie rings	second cohomology group for trivial group action of E8 on Z4#Generalized Baer Lie rings, second cohomology group for trivial group action of direct product of Z4 and Z2 on Z4#Generalized Baer Lie rings	second cohomology group for trivial group action of direct product of Z4 and Z4 on Z2
central product of D8 and Z8	38	$\Gamma_2g$	17	direct product of Z8 and V4	36	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring	generalized Baer correspondence between central product of D8 and Z8 and direct product of Z8 and V4	second cohomology group for trivial group action of V4 on Z8#Generalized Baer Lie rings	second cohomology group for trivial group action of E8 on Z4#Generalized Baer Lie rings	second cohomology group for trivial group action of direct product of Z4 and V4 on Z2
SmallGroup(32,24)	24	$\Gamma_2f$	16	direct product of Z4 and Z4 and Z2	21	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring	generalized Baer correspondence between SmallGroup(32,24) and direct product of Z4 and Z4 and Z2	second cohomology group for trivial group action of V4 on direct product of Z4 and Z2#Generalized Baer Lie rings	second cohomology group for trivial group action of direct product of Z4 and Z2 on Z4#Generalized Baer Lie rings	?
direct product of SmallGroup(16,13) and Z2	48	$\Gamma_2b$	10	direct product of E8 and Z4	45	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring	generalized Baer correspondence between direct product of SmallGroup(16,13) and Z2 and direct product of E8 on Z4	second cohomology group for trivial group action of V4 on direct product of Z4 and Z2#Generalized Baer Lie rings	second cohomology group for trivial group action of E8 on Z4#Generalized Baer Lie rings	?
SmallGroup(32,2)	2	$\Gamma_2h$	18	direct product of Z4 and Z4 and Z2	21	cocycle skew reversal generalization of Baer correspondence, the intermediate object being a class two near-Lie cring	generalized Baer correspondence between SmallGroup(32,2) and direct product of Z4 and Z4 and Z2	second cohomology group for trivial group action of direct product of Z4 and Z4 on Z2#Generalized Baer Lie rings	--	--
SmallGroup(32,33)	33	$\Gamma_4d$	41	direct product of Z4 and Z4 and Z2	21	mystery

For the first six of the cohomology perspective can be more compactly expressed as follows:

Cohomology	second cohomology group for trivial group action of V4 on direct product of Z4 and Z2#Generalized Baer Lie rings	second cohomology group for trivial group action of V4 on Z8#Generalized Baer Lie rings
second cohomology group for trivial group action of E8 on Z4#Generalized Baer Lie rings	direct product of SmallGroup(16,13) and Z2, direct product of M16 and Z2	central product of D8 and Z8
second cohomology group for trivial group action of direct product of Z4 and Z2 on Z4#Generalized Baer Lie rings	SmallGroup(32,24), semidirect product of Z8 and Z4 of M-type, direct product of M16 and Z2 (repeat)	M32

Grouping by abelian member

Of the seven abelian groups of order 32, five of them have non-abelian groups 1-isomorphic to them. The two missing ones are the obvious ones: cyclic group:Z32, on account of the fact that finite group having the same order statistics as a cyclic group is cyclic, and elementary abelian group:E32, on account of the fact that exponent two implies abelian.

Abelian member	GAP ID	Total number of members (excluding abelian member)	Other members	GAP IDs (in order of listing)	Hall-Senior symbols (in order of listing)	Hall-Senior numbers (in order of listing)
direct product of Z8 and Z4	3	1	semidirect product of Z8 and Z4 of M-type	4	$\Gamma_2i$	19
direct product of Z16 and Z2	16	1	M32	17	$\Gamma_2k$	22
direct product of Z4 and Z4 and Z2	21	3	SmallGroup(32,2), SmallGroup(32,24), SmallGroup(32,33)	2, 24, 33	$\Gamma_2h, \Gamma_2f, \Gamma_4d$	18, 16, 41
direct product of Z8 and V4	36	2	direct product of M16 and Z2, central product of D8 and Z8	37, 38	$\Gamma_2d, \Gamma_2g$	13, 17
direct product of E8 and Z4	45	1	direct product of SmallGroup(16,13) and Z2	48	$\Gamma_2b$	10
Total	--	8	--	--	--	--

Groupings that do not have any abelian member

These are groupings by 1-isomorphism where there are two or more members.

Members	GAP IDs (in order of listing)	Hall-Senior symbols (in order of listing)	Hall-Senior numbers (in order of listing)
direct product of Q8 and Z4 and SmallGroup(32,32)	26, 32	$\Gamma_2e_2, \Gamma_4c_3$	15, 40
direct product of SD16 and Z2 and central product of D16 and Z4	40, 42	$\Gamma_3a_2, \Gamma_3b$	24, 26
direct product of D8 and Z4, SmallGroup(32,30), SmallGroup(32,31)	25, 30, 31	$\Gamma_2e_1, \Gamma_4c_1, \Gamma_4c_2$	14, 38, 39
SmallGroup(32,27), generalized dihedral group for direct product of Z4 and Z4	27, 34	$\Gamma_4a_1, \Gamma_4a_2$	33, 34

Order statistics

FACTS TO CHECK AGAINST:

ORDER STATISTICS (cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots

1-ISOMORPHISM (cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic

Order statistics raw data

Note that because number of nth roots is a multiple of n, we see that the number of elements whose order is $1$ or $2$ is odd, while all the other numbers are even. The total number of $n^{th}$ roots is even for all $n = 2^k, k \ge 1$.

Group	Second part of GAP ID	Hall-Senior number	Number of elements of order 1	Number of elements of order 2	Number of elements of order 4	Number of elements of order 8	Number of elements of order 16	Number of elements of order 32
Cyclic group:Z32	1	7	1	1	2	4	8	16
SmallGroup(32,2)	2	18	1	7	24	0	0	0
Direct product of Z8 and Z4	3	5	1	3	12	16	0	0
SmallGroup(32,4)	4		1	3	12	16	0	0
SmallGroup(32,5)	5	20	1	7	8	16	0	0
Faithful semidirect product of E8 and Z4	6	46	1	11	20	0	0	0
SmallGroup(32,7)	7	47	1	11	4	16	0	0
SmallGroup(32,8)	8	48	1	3	12	16	0	0
SmallGroup(32,9)	9		1	11	12	8	0	0
SmallGroup(32,10)	10	28	1	3	20	8	0	0
Wreath product of Z4 and Z2	11	31	1	7	16	8	0	0
SmallGroup(32,12)	12	21	1	3	12	16	0	0
SmallGroup(32,13)	13		1	3	20	8	0	0
SmallGroup(32,14)	14		1	3	20	8	0	0
SmallGroup(32,15)	15	32	1	3	4	24	0	0
Direct product of Z16 and Z2	16	6	1	3	4	8	16	0
M32	17	22	1	3	4	8	16	0
Dihedral group:D32	18	49	1	17	2	4	8	0
Semidihedral group:SD32	19	50	1	9	10	4	8	0
Generalized quaternion group:Q32	20	51	1	1	18	4	8	0
Direct product of Z4 and Z4 and Z2	21	3	1	7	24	0	0	0
Direct product of SmallGroup(16,3) and Z2	22	11	1	15	16	0	0	0
Direct product of SmallGroup(16,4) and Z2	23	12	1	7	24	0	0	0
SmallGroup(32,24)	24	16	1	7	24	0	0	0
Direct product of D8 and Z4	25	14	1	11	20	0	0	0
Direct product of Q8 and Z4	26	15	1	3	28	0	0	0
SmallGroup(32,27)	27	33	1	19	12	0	0	0
SmallGroup(32,28)	28	36	1	15	16	0	0	0
SmallGroup(32,29)	29	37	1	7	24	0	0	0
SmallGroup(32,30)	30	38	1	11	20	0	0	0
SmallGroup(32,31)	31	39	1	11	20	0	0	0
SmallGroup(32,32)	32	40	1	3	28	0	0	0
SmallGroup(32,33)	33	41	1	7	24	0	0	0
Generalized dihedral group for direct product of Z4 and Z4	34	34	1	19	12	0	0	0
SmallGroup(32,35)	35	35	1	3	28	0	0	0
Direct product of Z8 and V4	36	4	1	7	8	16	0	0
Direct product of M16 and Z2	37	13	1	7	8	16	0	0
SmallGroup(32,38)	38	17	1	7	8	16	0	0
Direct product of D16 and Z2	39	23	1	19	4	8	0	0
Direct product of SD16 and Z2	40	24	1	11	12	8	0	0
SmallGroup(32,41)	41	25	1	3	20	8	0	0
SmallGroup(32,42)	42		1	11	12	8	0	0
Holomorph of Z8	43	44	1	15	8	8	0	0
SmallGroup(32,44)	44	45	1	7	16	8	0	0
Direct product of E8 and Z4	45	2	1	15	16	0	0	0
Direct product of D8 and V4	46	8	1	23	8	0	0	0
Direct product of Q8 and V4	47	9	1	7	24	0	0	0
Direct product of SmallGroup(16,13) and Z2	48	10	1	15	16	0	0	0
Inner holomorph of D8	49	42	1	19	12	0	0	0
SmallGroup(32,50)	50	43	1	11	20	0	0	0
Elementary abelian group:E32	51	1	1	31	0	0	0	0

Here is the GAP code to generate these order statistics:[SHOW MORE]

gap> F := List(AllSmallGroups(32),G -> List(Set(G),Order));;
gap> K := List(F,L->[Length(Filtered(L,x -> x = 1)),
> Length(Filtered(L,x -> x = 2)),Length(Filtered(L,x -> x = 4)),
> Length(Filtered(L,x -> x = 8)),Length(Filtered(L,x->x=16)),Length(Filtered(L,x ->x=32))]);;
gap> M := List([1..51], i ->[i,K[i]]);

Here is GAP's output:

[ [ 1, [ 1, 1, 2, 4, 8, 16 ] ], [ 2, [ 1, 7, 24, 0, 0, 0 ] ], [ 3, [ 1, 3, 12, 16, 0, 0 ] ], [ 4, [ 1, 3, 12, 16, 0, 0 ] ], [ 5, [ 1, 7, 8, 16, 0, 0 ] ],
  [ 6, [ 1, 11, 20, 0, 0, 0 ] ], [ 7, [ 1, 11, 4, 16, 0, 0 ] ], [ 8, [ 1, 3, 12, 16, 0, 0 ] ], [ 9, [ 1, 11, 12, 8, 0, 0 ] ],
  [ 10, [ 1, 3, 20, 8, 0, 0 ] ], [ 11, [ 1, 7, 16, 8, 0, 0 ] ], [ 12, [ 1, 3, 12, 16, 0, 0 ] ], [ 13, [ 1, 3, 20, 8, 0, 0 ] ],
  [ 14, [ 1, 3, 20, 8, 0, 0 ] ], [ 15, [ 1, 3, 4, 24, 0, 0 ] ], [ 16, [ 1, 3, 4, 8, 16, 0 ] ], [ 17, [ 1, 3, 4, 8, 16, 0 ] ],
  [ 18, [ 1, 17, 2, 4, 8, 0 ] ], [ 19, [ 1, 9, 10, 4, 8, 0 ] ], [ 20, [ 1, 1, 18, 4, 8, 0 ] ], [ 21, [ 1, 7, 24, 0, 0, 0 ] ],
  [ 22, [ 1, 15, 16, 0, 0, 0 ] ], [ 23, [ 1, 7, 24, 0, 0, 0 ] ], [ 24, [ 1, 7, 24, 0, 0, 0 ] ], [ 25, [ 1, 11, 20, 0, 0, 0 ] ],
  [ 26, [ 1, 3, 28, 0, 0, 0 ] ], [ 27, [ 1, 19, 12, 0, 0, 0 ] ], [ 28, [ 1, 15, 16, 0, 0, 0 ] ], [ 29, [ 1, 7, 24, 0, 0, 0 ] ],
  [ 30, [ 1, 11, 20, 0, 0, 0 ] ], [ 31, [ 1, 11, 20, 0, 0, 0 ] ], [ 32, [ 1, 3, 28, 0, 0, 0 ] ], [ 33, [ 1, 7, 24, 0, 0, 0 ] ],
  [ 34, [ 1, 19, 12, 0, 0, 0 ] ], [ 35, [ 1, 3, 28, 0, 0, 0 ] ], [ 36, [ 1, 7, 8, 16, 0, 0 ] ], [ 37, [ 1, 7, 8, 16, 0, 0 ] ],
  [ 38, [ 1, 7, 8, 16, 0, 0 ] ], [ 39, [ 1, 19, 4, 8, 0, 0 ] ], [ 40, [ 1, 11, 12, 8, 0, 0 ] ], [ 41, [ 1, 3, 20, 8, 0, 0 ] ],
  [ 42, [ 1, 11, 12, 8, 0, 0 ] ], [ 43, [ 1, 15, 8, 8, 0, 0 ] ], [ 44, [ 1, 7, 16, 8, 0, 0 ] ], [ 45, [ 1, 15, 16, 0, 0, 0 ] ],
  [ 46, [ 1, 23, 8, 0, 0, 0 ] ], [ 47, [ 1, 7, 24, 0, 0, 0 ] ], [ 48, [ 1, 15, 16, 0, 0, 0 ] ], [ 49, [ 1, 19, 12, 0, 0, 0 ] ],
  [ 50, [ 1, 11, 20, 0, 0, 0 ] ], [ 51, [ 1, 31, 0, 0, 0, 0 ] ] ]

Here now are the cumulative order statistics:

Group	Second part of GAP ID	Hall-Senior number	Number of 1st roots	Number of 2nd roots	Number of 4th roots	Number of 8th roots	Number of 16th roots	Number of 32nd roots
Cyclic group:Z32	1	7	1	2	4	8	16	32
SmallGroup(32,2)	2	18	1	8	32	32	32	32
Direct product of Z8 and Z4	3	5	1	4	16	32	32	32
SmallGroup(32,4)	4		1	4	16	32	32	32
SmallGroup(32,5)	5	20	1	8	16	32	32	32
Faithful semidirect product of E8 and Z4	6	46	1	12	32	32	32	32
SmallGroup(32,7)	7	47	1	12	16	32	32	32
SmallGroup(32,8)	8	48	1	4	16	32	32	32
SmallGroup(32,9)	9		1	12	24	32	32	32
SmallGroup(32,10)	10	28	1	4	24	32	32	32
Wreath product of Z4 and Z2	11	31	1	8	24	32	32	32
SmallGroup(32,12)	12	21	1	4	16	32	32	32
SmallGroup(32,13)	13		1	4	24	32	32	32
SmallGroup(32,14)	14		1	4	24	32	32	32
SmallGroup(32,15)	15	32	1	4	8	32	32	32
Direct product of Z16 and Z2	16	6	1	4	8	16	32	32
M32	17	22	1	4	8	16	32	32
Dihedral group:D32	18	49	1	18	20	24	32	32
Semidihedral group:SD32	19	50	1	10	20	24	32	32
Generalized quaternion group:Q32	20	51	1	2	20	24	32	32
Direct product of Z4 and Z4 and Z2	21	3	1	8	32	32	32	32
Direct product of SmallGroup(16,3) and Z2	22	11	1	16	32	32	32	32
Direct product of SmallGroup(16,4) and Z2	23	12	1	8	32	32	32	32
SmallGroup(32,24)	24	16	1	8	32	32	32	32
Direct product of D8 and Z4	25	14	1	12	32	32	32	32
Direct product of Q8 and Z4	26	15	1	4	32	32	32	32
SmallGroup(32,27)	27	33	1	20	32	32	32	32
SmallGroup(32,28)	28	36	1	16	32	32	32	32
SmallGroup(32,29)	29	37	1	8	32	32	32	32
SmallGroup(32,30)	30	38	1	12	32	32	32	32
SmallGroup(32,31)	31	39	1	12	32	32	32	32
SmallGroup(32,32)	32	40	1	4	32	32	32	32
SmallGroup(32,33)	33	41	1	8	32	32	32	32
Generalized dihedral group for direct product of Z4 and Z4	34	34	1	20	32	32	32	32
SmallGroup(32,35)	35	35	1	4	32	32	32	32
Direct product of Z8 and V4	36	4	1	8	16	32	32	32
Direct product of M16 and Z2	37	13	1	8	16	32	32	32
SmallGroup(32,38)	38	17	1	8	16	32	32	32
Direct product of D16 and Z2	39	23	1	20	24	32	32	32
Direct product of SD16 and Z2	40	24	1	12	24	32	32	32
SmallGroup(32,41)	41	25	1	4	24	32	32	32
SmallGroup(32,42)	42		1	12	24	32	32	32
Holomorph of Z8	43	44	1	16	24	32	32	32
SmallGroup(32,44)	44	45	1	8	24	32	32	32
Direct product of E8 and Z4	45	2	1	16	32	32	32	32
Direct product of D8 and V4	46	8	1	24	32	32	32	32
Direct product of Q8 and V4	47	9	1	8	32	32	32	32
Direct product of SmallGroup(16,13) and Z2	48	10	1	16	32	32	32	32
Inner holomorph of D8	49	42	1	20	32	32	32	32
SmallGroup(32,50)	50	43	1	12	32	32	32	32
Elementary abelian group:E32	51	1	1	32	32	32	32	32

Here are the GAP commands to generate the cumulative order statistics: [SHOW MORE]

gap> F := List(AllSmallGroups(32),G -> List(Set(G),Order));;
gap> J := List(F,L->[Length(Filtered(L,x -> x <= 1)),
> Length(Filtered(L,x -> x <= 2)),Length(Filtered(L,x -> x <= 4)),
> Length(Filtered(L,x -> x <= 8)),Length(Filtered(L,x->x<=16)),Length(Filtered(L,x ->x<=32))]);;
gap> N := List([1..51], i ->[i,J[i]]);

Here is GAP's output:

[ [ 1, [ 1, 2, 4, 8, 16, 32 ] ], [ 2, [ 1, 8, 32, 32, 32, 32 ] ], [ 3, [ 1, 4, 16, 32, 32, 32 ] ], [ 4, [ 1, 4, 16, 32, 32, 32 ] ],
  [ 5, [ 1, 8, 16, 32, 32, 32 ] ], [ 6, [ 1, 12, 32, 32, 32, 32 ] ], [ 7, [ 1, 12, 16, 32, 32, 32 ] ], [ 8, [ 1, 4, 16, 32, 32, 32 ] ],
  [ 9, [ 1, 12, 24, 32, 32, 32 ] ], [ 10, [ 1, 4, 24, 32, 32, 32 ] ], [ 11, [ 1, 8, 24, 32, 32, 32 ] ], [ 12, [ 1, 4, 16, 32, 32, 32 ] ],
  [ 13, [ 1, 4, 24, 32, 32, 32 ] ], [ 14, [ 1, 4, 24, 32, 32, 32 ] ], [ 15, [ 1, 4, 8, 32, 32, 32 ] ], [ 16, [ 1, 4, 8, 16, 32, 32 ] ],
  [ 17, [ 1, 4, 8, 16, 32, 32 ] ], [ 18, [ 1, 18, 20, 24, 32, 32 ] ], [ 19, [ 1, 10, 20, 24, 32, 32 ] ], [ 20, [ 1, 2, 20, 24, 32, 32 ] ],
  [ 21, [ 1, 8, 32, 32, 32, 32 ] ], [ 22, [ 1, 16, 32, 32, 32, 32 ] ], [ 23, [ 1, 8, 32, 32, 32, 32 ] ], [ 24, [ 1, 8, 32, 32, 32, 32 ] ],
  [ 25, [ 1, 12, 32, 32, 32, 32 ] ], [ 26, [ 1, 4, 32, 32, 32, 32 ] ], [ 27, [ 1, 20, 32, 32, 32, 32 ] ], [ 28, [ 1, 16, 32, 32, 32, 32 ] ],
  [ 29, [ 1, 8, 32, 32, 32, 32 ] ], [ 30, [ 1, 12, 32, 32, 32, 32 ] ], [ 31, [ 1, 12, 32, 32, 32, 32 ] ], [ 32, [ 1, 4, 32, 32, 32, 32 ] ],
  [ 33, [ 1, 8, 32, 32, 32, 32 ] ], [ 34, [ 1, 20, 32, 32, 32, 32 ] ], [ 35, [ 1, 4, 32, 32, 32, 32 ] ], [ 36, [ 1, 8, 16, 32, 32, 32 ] ],
  [ 37, [ 1, 8, 16, 32, 32, 32 ] ], [ 38, [ 1, 8, 16, 32, 32, 32 ] ], [ 39, [ 1, 20, 24, 32, 32, 32 ] ], [ 40, [ 1, 12, 24, 32, 32, 32 ] ],
  [ 41, [ 1, 4, 24, 32, 32, 32 ] ], [ 42, [ 1, 12, 24, 32, 32, 32 ] ], [ 43, [ 1, 16, 24, 32, 32, 32 ] ], [ 44, [ 1, 8, 24, 32, 32, 32 ] ],
  [ 45, [ 1, 16, 32, 32, 32, 32 ] ], [ 46, [ 1, 24, 32, 32, 32, 32 ] ], [ 47, [ 1, 8, 32, 32, 32, 32 ] ], [ 48, [ 1, 16, 32, 32, 32, 32 ] ],
  [ 49, [ 1, 20, 32, 32, 32, 32 ] ], [ 50, [ 1, 12, 32, 32, 32, 32 ] ], [ 51, [ 1, 32, 32, 32, 32, 32 ] ] ]

Equivalence classes based on order statistics

Here, we discuss the equivalence classes of groups of order 32 up to being order statistics-equivalent finite groups and up to the stronger notion of being 1-isomorphic groups (which means there is a bijection that restricts to isomorphisms on cyclic subgroups). See also order statistics-equivalent not implies 1-isomorphic.

Order statistics	Order statistics (cumulative)	Number of groups	Number of equivalence classes up to 1-isomorphism	Members of first equivalence class	Members of second equivalence class	Members of third equivalence class	Members of fourth equivalence class	Abelian group with these order statistics?	Cumulative order statistics all powers of 2?
1,1,2,4,8,16	1,2,4,8,16,32	1	1	cyclic group:Z32 (ID:1)				Yes	Yes
1,1,18,4,8,0	1,2,20,24,32,32	1	1	generalized quaternion group:Q32 (ID:20)				No	No
1,3,4,8,16,0	1,4,8,16,32,32	2	1	direct product of Z16 and Z2 (ID:16) and M32 (ID:17)				Yes	Yes
1,3,4,24,0,0	1,4,8,32,32,32	1	1	SmallGroup(32,15) (ID:15)				No	Yes
1,3,12,16,0,0	1,4,16,32,32,32	4	3	direct product of Z8 and Z4 (ID:3) and semidirect product of Z8 and Z4 of M-type (ID:4)	SmallGroup(32,8)	SmallGroup(32,12)		Yes	No
1,3,20,8,0,0	1,4,24,32,32,32	4	4	SmallGroup(32,10) (ID:10)	semidirect product of Z8 and Z4 of dihedral type (ID:14)	semidirect product of Z8 and Z4 of semidihedral type (ID:13)	SmallGroup(32,41)	No	No
1,3,28,0,0,0	1,4,32,32,32,32	3	2	direct product of Q8 and Z4 (ID:26) and SmallGroup(32,32) (ID:32)	SmallGroup(32,35)			No	Yes
1,7,8,16,0,0	1,8,16,32,32,32	4	2	direct product of Z8 and V4 (ID:36), direct product of M16 and Z2 (ID:37), SmallGroup(32,38) (ID:38)	SmallGroup(32,5)			Yes	Yes
1,7,16,8,0,0	1,8,24,32,32,32	2	2	SmallGroup(32,11) (ID:11)	SmallGroup(32,44) (ID:44)			No	Yes
1,7,24,0,0,0	1,8,32,32,32,32	7	3	direct product of Z4 and Z4 and Z2 (ID:21), SmallGroup(32,2) (ID:2), SmallGroup(32,24) (ID:24), SmallGroup(32,33) (ID:33)	direct product of SmallGroup(16,4) and Z2 (ID:23) and SmallGroup(32,29) (ID:29)	SmallGroup(32,47)		Yes	Yes
1,9,10,4,8,0	1,10,20,24,32,32	1	1	semidihedral group:SD32 (ID:19)				No	No
1,11,4,16,0,0	1,12,16,32,32,32	1	1	SmallGroup(32,7) (ID:7)				No	No
1,11,12,8,0,0	1,12,24,32,32,32	3	2	direct product of SD16 and Z2 (ID:40) and central product of D16 and Z4 (ID:42)	SmallGroup(32,9)			No	No
1,11,20,0,0,0	1,12,32,32,32,32	5	3	direct product of D8 and Z4 (ID:25), SmallGroup(32,30) (ID:30), SmallGroup(32,31) (ID:31)	faithful semidirect product of E8 and Z4 (ID:6)	central product of D8 and Q8 (ID:50)		No	No
1,15,8,8,0,0	1,16,24,32,32,32	1	1	holomorph of Z8 (ID:43)				No	No
1,15,16,0,0,0	1,16,32,32,32,32	4	2	direct product of E8 and Z4 (ID:45) and direct product of SmallGroup(16,13) and Z2 (ID:48)	direct product of SmallGroup(16,3) and Z2 (ID: 22) and SmallGroup(32,28) (ID:28)			Yes	Yes
1,17,2,4,8,0	1,18,20,24,32,32	1	1	dihedral group:D32 (ID:18)				No	No
1,19,4,8,0,0	1,20,24,32,32,32	1	1	direct product of D16 and Z2 (ID:39)				No	No
1,19,12,0,0,0	1,20,32,32,32,32	3	?	SmallGroup(32,27) (ID:27) and generalized dihedral group for direct product of Z4 and Z4 (ID:34)	inner holomorph of D8 (ID:49)			No	No
1,23,8,0,0,0	1,24,32,32,32,32	1	1	direct product of D8 and V4 (ID:46)				No	No
1,31,0,0,0,0	1,32,32,32,32,32	1	1	elementary abelian group:E32 (ID:51)				Yes	Yes

Here is the GAP code to sort all groups of order 32 by equivalence classes:[SHOW MORE]

gap> F := List(AllSmallGroups(32),G -> List(Set(G),Order));;
gap> K := List(F,L->[Length(Filtered(L,x -> x = 1)),
> Length(Filtered(L,x -> x = 2)),Length(Filtered(L,x -> x = 4)),
> Length(Filtered(L,x -> x = 8)),Length(Filtered(L,x -> x = 16)), Length(Filtered(L,x -> x = 32))]);;
gap> M := List([1..51], i -> [K[i],i]);;
gap> S := SortedList(M);

Here is GAP's output:

[ [ [ 1, 1, 2, 4, 8, 16 ], 1 ], [ [ 1, 1, 18, 4, 8, 0 ], 20 ], [ [ 1, 3, 4, 8, 16, 0 ], 16 ], [ [ 1, 3, 4, 8, 16, 0 ], 17 ],
  [ [ 1, 3, 4, 24, 0, 0 ], 15 ], [ [ 1, 3, 12, 16, 0, 0 ], 3 ], [ [ 1, 3, 12, 16, 0, 0 ], 4 ], [ [ 1, 3, 12, 16, 0, 0 ], 8 ],
  [ [ 1, 3, 12, 16, 0, 0 ], 12 ], [ [ 1, 3, 20, 8, 0, 0 ], 10 ], [ [ 1, 3, 20, 8, 0, 0 ], 13 ], [ [ 1, 3, 20, 8, 0, 0 ], 14 ],
  [ [ 1, 3, 20, 8, 0, 0 ], 41 ], [ [ 1, 3, 28, 0, 0, 0 ], 26 ], [ [ 1, 3, 28, 0, 0, 0 ], 32 ], [ [ 1, 3, 28, 0, 0, 0 ], 35 ],
  [ [ 1, 7, 8, 16, 0, 0 ], 5 ], [ [ 1, 7, 8, 16, 0, 0 ], 36 ], [ [ 1, 7, 8, 16, 0, 0 ], 37 ], [ [ 1, 7, 8, 16, 0, 0 ], 38 ],
  [ [ 1, 7, 16, 8, 0, 0 ], 11 ], [ [ 1, 7, 16, 8, 0, 0 ], 44 ], [ [ 1, 7, 24, 0, 0, 0 ], 2 ], [ [ 1, 7, 24, 0, 0, 0 ], 21 ],
  [ [ 1, 7, 24, 0, 0, 0 ], 23 ], [ [ 1, 7, 24, 0, 0, 0 ], 24 ], [ [ 1, 7, 24, 0, 0, 0 ], 29 ], [ [ 1, 7, 24, 0, 0, 0 ], 33 ],
  [ [ 1, 7, 24, 0, 0, 0 ], 47 ], [ [ 1, 9, 10, 4, 8, 0 ], 19 ], [ [ 1, 11, 4, 16, 0, 0 ], 7 ], [ [ 1, 11, 12, 8, 0, 0 ], 9 ],
  [ [ 1, 11, 12, 8, 0, 0 ], 40 ], [ [ 1, 11, 12, 8, 0, 0 ], 42 ], [ [ 1, 11, 20, 0, 0, 0 ], 6 ], [ [ 1, 11, 20, 0, 0, 0 ], 25 ],
  [ [ 1, 11, 20, 0, 0, 0 ], 30 ], [ [ 1, 11, 20, 0, 0, 0 ], 31 ], [ [ 1, 11, 20, 0, 0, 0 ], 50 ], [ [ 1, 15, 8, 8, 0, 0 ], 43 ],
  [ [ 1, 15, 16, 0, 0, 0 ], 22 ], [ [ 1, 15, 16, 0, 0, 0 ], 28 ], [ [ 1, 15, 16, 0, 0, 0 ], 45 ], [ [ 1, 15, 16, 0, 0, 0 ], 48 ],
  [ [ 1, 17, 2, 4, 8, 0 ], 18 ], [ [ 1, 19, 4, 8, 0, 0 ], 39 ], [ [ 1, 19, 12, 0, 0, 0 ], 27 ], [ [ 1, 19, 12, 0, 0, 0 ], 34 ],
  [ [ 1, 19, 12, 0, 0, 0 ], 49 ], [ [ 1, 23, 8, 0, 0, 0 ], 46 ], [ [ 1, 31, 0, 0, 0, 0 ], 51 ] ]