Groups of order 32

Statistics at a glance

Quantity	Value
Number of groups up to isomorphism	51
Number of abelian groups up to isomorphism	7
Number of groups of class exactly two up to isomorphism	26
Number of groups of class exactly three up to isomorphism	15
Number of groups of class exactly four up to isomorphism	3

The list

Note there's an ambiguity that makes the table below incomplete: the Hall-Senior numbers of groups with GAP IDs 13 and 14 are 29 and 30 (symbol ${\mathrm{\Gamma }}_3{d}_1$ and ${\mathrm{\Gamma }}_3{d}_2$ respectively) but it's not yet clear which GAP ID corresponds to which Hall-Senior number.

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
Cyclic group:Z32	1	7	$(5)$	1
SmallGroup(32,2)	2	18	${\mathrm{\Gamma }}_{2}h$	2
Direct product of Z8 and Z4	3	5	$(32)$	1
Semidirect product of Z8 and Z4 of M-type	4	19	${\mathrm{\Gamma }}_{2}i$	2
SmallGroup(32,5)	5	20	${\mathrm{\Gamma }}_2{j}_1$	2
Faithful semidirect product of E8 and Z4	6	46	${\mathrm{\Gamma }}_7{a}_1$	3
SmallGroup(32,7)	7	47	${\mathrm{\Gamma }}_7{a}_2$	3
SmallGroup(32,8)	8	48	${\mathrm{\Gamma }}_7{a}_3$	3
SmallGroup(32,9)	9	27	${\mathrm{\Gamma }}_3{c}_1$	3
SmallGroup(32,10)	10	28	${\mathrm{\Gamma }}_3{c}_2$	3
Wreath product of Z4 and Z2	11	31	${\mathrm{\Gamma }}_{3}e$	3
SmallGroup(32,12)	12	21	${\mathrm{\Gamma }}_2{j}_2$	2
Semidirect product of Z8 and Z4 of semidihedral type	13			3
Semidirect product of Z8 and Z4 of dihedral type	14			3
SmallGroup(32,15)	15	32	${\mathrm{\Gamma }}_{3}f$	3
Direct product of Z16 and Z2	16	6	$(41)$	1
M32	17	22	${\mathrm{\Gamma }}_{2}k$	2
Dihedral group:D32	18	49	${\mathrm{\Gamma }}_8{a}_1$	4
Semidihedral group:SD32	19	50	${\mathrm{\Gamma }}_8{a}_2$	4
Generalized quaternion group:Q32	20	51	${\mathrm{\Gamma }}_8{a}_3$	4
Direct product of Z4 and Z4 and Z2	21	3	$(2^{2}1)$	1
Direct product of SmallGroup(16,3) and Z2	22	11	${\mathrm{\Gamma }}_2{c}_1$	2
Direct product of SmallGroup(16,4) and Z2	23	12	${\mathrm{\Gamma }}_2{c}_2$	2
SmallGroup(32,24)	24	16	${\mathrm{\Gamma }}_{2}f$	2
Direct product of D8 and Z4	25	14	${\mathrm{\Gamma }}_2{e}_1$	2
Direct product of Q8 and Z4	26	15	${\mathrm{\Gamma }}_2{e}_2$	2
SmallGroup(32,27)	27	33	${\mathrm{\Gamma }}_4{a}_1$	2
SmallGroup(32,28)	28	36	${\mathrm{\Gamma }}_4{b}_1$	2
SmallGroup(32,29)	29	37	${\mathrm{\Gamma }}_4{b}_2$	2
SmallGroup(32,30)	30	38	${\mathrm{\Gamma }}_4{c}_1$	2
SmallGroup(32,31)	31	39	${\mathrm{\Gamma }}_4{c}_2$	2
SmallGroup(32,32)	32	40	${\mathrm{\Gamma }}_4{c}_3$	2
SmallGroup(32,33)	33	41	${\mathrm{\Gamma }}_{4}d$	2
Generalized dihedral group for direct product of Z4 and Z4	34	34	${\mathrm{\Gamma }}_4{a}_2$	2
SmallGroup(32,35)	35	35	${\mathrm{\Gamma }}_4{a}_3$	2
Direct product of Z8 and V4	36	4	$(31^2)$	1
Direct product of M16 and Z2	37	13	${\mathrm{\Gamma }}_{2}d$	2
Central product of D8 and Z8	38	17	${\mathrm{\Gamma }}_{2}g$	2
Direct product of D16 and Z2	39	23	${\mathrm{\Gamma }}_3{a}_1$	3
Direct product of SD16 and Z2	40	24	${\mathrm{\Gamma }}_3{a}_2$	3
Direct product of Q16 and Z2	41	25	${\mathrm{\Gamma }}_3{a}_3$	3
Central product of D16 and Z4	42	26	${\mathrm{\Gamma }}_{3}b$	3
Holomorph of Z8	43	44	${\mathrm{\Gamma }}_6{a}_1$	3
SmallGroup(32,44)	44	45	${\mathrm{\Gamma }}_6{a}_2$	3
Direct product of E8 and Z4	45	2	$(21^3)$	1
Direct product of D8 and V4	46	8	${\mathrm{\Gamma }}_2{a}_1$	2
Direct product of Q8 and V4	47	9	${\mathrm{\Gamma }}_2{a}_2$	2
Direct product of SmallGroup(16,13) and Z2	48	10	${\mathrm{\Gamma }}_{2}b$	2
Inner holomorph of D8	49	42	${\mathrm{\Gamma }}_5{a}_1$	2
Central product of D8 and Q8	50	43	${\mathrm{\Gamma }}_5{a}_2$	2
Elementary abelian group:E32	51	1	$(1^5)$	1

Arithmetic functions

Summary information

Here, the rows are arithmetic functions that take values between $0$ and $5$, and the columns give the possible values of these functions. The entry in each cell is the number of isomorphism classes of groups for which the row arithmetic function takes the column value. Note that all the row value sums must equal $51$.

Arithmetic function	Value 0	Value 1	Value 2	Value 3	Value 4	Value 5
prime-base logarithm of exponent	0	1	23	21	5	1
Frattini length	0	1	23	21	5	1
nilpotency class	0	7	26	15	3	0
derived length	0	7	44	0	0	0
minimum size of generating set	0	1	19	24	6	1
rank of a p-group	0	2	21	23	4	1
normal rank of a p-group	0	4	23	19	4	1
characteristic rank of a p-group	0	7	26	14	3	1

Families and classification

Isocliny, or Hall-Senior families

Family name	Isomorphism class of inner automorphism group	Isomorphism class of derived subgroup	Number of members	Nilpotency class	Members	Second part of GAP ID of members (sorted ascending)	Hall-Senior numbers of members (sorted ascending)
${\mathrm{\Gamma }}_1$	trivial group	trivial group	7	1	all 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 Z4 and E8, elementary abelian group:E32	1,3,16,21,36,45,51	1-7
${\mathrm{\Gamma }}_2$	Klein four-group	cyclic group:Z2	15	2	direct product of D8 and V4, direct product of Q8 and V4, direct product of SmallGroup(16,13) and Z2, direct product of SmallGroup(16,3) and Z2, direct product of SmallGroup(16,4) and Z2, direct product of M16 and Z2, direct product of D8 and Z4, direct product of Q8 and Z4, SmallGroup(32,24), central product of D8 and Z8, SmallGroup(32,2), SmallGroup(32,5), SmallGroup(32,12), SmallGroup(32,12), M32, semidirect product of Z8 and Z4 of M-type	2,4,5,12,17,22,23,24,25,26,37,38,46,47,48	8-22
${\mathrm{\Gamma }}_3$	dihedral group:D8	cyclic group:Z4	10	3	direct product of D16 and Z2, direct product of SD16 and Z2, direct product of Q16 and Z2, central product of D16 and Z4, semidirect product of Z8 and Z4 of dihedral type, semidirect product of Z8 and Z4 of semidihedral type, SmallGroup(32,9), SmallGroup(32,10), wreath product of Z4 and Z2, SmallGroup(32,15)	9,10,11,13,14,15,39,40,41,42	23-32
${\mathrm{\Gamma }}_4$	elementary abelian group:E8	Klein four-group	9	2	SmallGroup(32,27), SmallGroup(32,28), SmallGroup(32,29), SmallGroup(32,30), SmallGroup(32,31), SmallGroup(32,32), SmallGroup(32,33), generalized dihedral group for direct product of Z4 and Z4, SmallGroup(32,35)	27-35	33-41
${\mathrm{\Gamma }}_5$	elementary abelian group:E16	cyclic group:Z2	2	2	inner holomorph of D8, central product of D8 and Q8	49, 50	42, 43
${\mathrm{\Gamma }}_6$	direct product of D8 and Z2	cyclic group:Z2	2	3	holomorph of Z8, SmallGroup(32,44)	43,44	44,45
${\mathrm{\Gamma }}_7$	SmallGroup(16,3)	Klein four-group	3	3	faithful semidirect product of E8 and Z4, SmallGroup(32,7), SmallGroup(32,8)	6-8	46-48
${\mathrm{\Gamma }}_8$	dihedral group:D16	cyclic group:Z8	3	4	dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32	18-20	49-51

Hall-Senior genus

Genus name	Members	Second part of GAP ID of members	Hall-Senior numbers of members
$(1^5)$	elementary abelian group:E32	51	1
$(21^3)$	direct product of E8 and Z4	45	2
$(2^{2}1)$	direct product of Z4 and Z4 and Z2	21	3
$(31^2)$	direct product of Z8 and V4	36	4
$(32)$	direct product of Z8 and Z4	3	5
$(41)$	direct product of Z16 and Z2	16	6
$(5)$	cyclic group:Z32	1	7
${\mathrm{\Gamma }}_{2}a$	direct product of D8 and V4, direct product of Q8 and V4	46,47	8,9
${\mathrm{\Gamma }}_{2}b$	direct product of SmallGroup(16,13) and Z2	48	10
${\mathrm{\Gamma }}_{2}c$	direct product of SmallGroup(16,3) and Z2, direct product of SmallGroup(16,4) and Z2	22,23	11,12
${\mathrm{\Gamma }}_{2}d$	direct product of M16 and Z2	37	13
${\mathrm{\Gamma }}_{2}e$	direct product of D8 and Z4, direct product of Q8 and Z4	25, 26	14, 15
${\mathrm{\Gamma }}_{2}f$	SmallGroup(32,24)	24	16
${\mathrm{\Gamma }}_{2}g$	central product of D8 and Z8	38	17
${\mathrm{\Gamma }}_{2}h$	SmallGroup(32,2)	2	18
${\mathrm{\Gamma }}_{2}i$	semidirect product of Z8 and Z4 of M-type	4	19
${\mathrm{\Gamma }}_{2}j$	SmallGroup(32,5), SmallGroup(32,12)	5, 12	20, 21
${\mathrm{\Gamma }}_{2}k$	M32	17	22
${\mathrm{\Gamma }}_{3}a$	direct product of D16 and Z2, direct product of SD16 and Z2, direct product of Q16 and Z2	39-41	23-25
${\mathrm{\Gamma }}_{3}b$	central product of D8 and Z8	42	26

Element structure

Further information: element structure of groups of order 32

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

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		1	2	4	8	16	32
SmallGroup(32,2)	2		1	8	32	32	32	32
Direct product of Z8 and Z4	3		1	4	16	32	32	32
Semidirect product of Z8 and Z4 of M-type	4		1	4	16	32	32	32
SmallGroup(32,5)	5		1	8	16	32	32	32
Faithful semidirect product of E8 and Z4	6		1	12	32	32	32	32
SmallGroup(32,7)	7		1	12	16	32	32	32
SmallGroup(32,8)	8		1	4	16	32	32	32
SmallGroup(32,9)	9		1	12	24	32	32	32
SmallGroup(32,10)	10		1	4	24	32	32	32
Wreath product of Z4 and Z2	11		1	8	24	32	32	32
SmallGroup(32,12)	12		1	4	16	32	32	32
Semidirect product of Z8 and Z4 of semidihedral type	13		1	4	24	32	32	32
Semidirect product of Z8 and Z4 of dihedral type	14		1	4	24	32	32	32
SmallGroup(32,15)	15		1	4	8	32	32	32
Direct product of Z16 and Z2	16		1	4	8	16	32	32
M32	17		1	4	8	16	32	32
Dihedral group:D32	18		1	18	20	24	32	32
Semidihedral group:SD32	19		1	10	20	24	32	32
Generalized quaternion group:Q32	20		1	2	20	24	32	32
Direct product of Z4 and Z4 and Z2	21		1	8	32	32	32	32
Direct product of SmallGroup(16,3) and Z2	22		1	16	32	32	32	32
Direct product of SmallGroup(16,4) and Z2	23		1	8	32	32	32	32
SmallGroup(32,24)	24		1	8	32	32	32	32
Direct product of D8 and Z4	25		1	12	32	32	32	32
Direct product of Q8 and Z4	26		1	4	32	32	32	32
SmallGroup(32,27)	27		1	20	32	32	32	32
SmallGroup(32,28)	28		1	16	32	32	32	32
SmallGroup(32,29)	29		1	8	32	32	32	32
SmallGroup(32,30)	30		1	12	32	32	32	32
SmallGroup(32,31)	31		1	12	32	32	32	32
SmallGroup(32,32)	32		1	4	32	32	32	32
SmallGroup(32,33)	33		1	8	32	32	32	32
Generalized dihedral group for direct product of Z4 and Z4	34		1	20	32	32	32	32
SmallGroup(32,35)	35		1	4	32	32	32	32
Direct product of Z8 and V4	36		1	8	16	32	32	32
Direct product of M16 and Z2	37		1	8	16	32	32	32
SmallGroup(32,38)	38		1	8	16	32	32	32
Direct product of D16 and Z2	39		1	20	24	32	32	32
Direct product of SD16 and Z2	40		1	12	24	32	32	32
SmallGroup(32,41)	41		1	4	24	32	32	32
SmallGroup(32,42)	42		1	12	24	32	32	32
Holomorph of Z8	43		1	16	24	32	32	32
SmallGroup(32,44)	44		1	8	24	32	32	32
Direct product of E8 and Z4	45		1	16	32	32	32	32
Direct product of D8 and V4	46		1	24	32	32	32	32
Direct product of Q8 and V4	47		1	8	32	32	32	32
Direct product of SmallGroup(16,13) and Z2	48		1	16	32	32	32	32
Inner holomorph of D8	49		1	20	32	32	32	32
SmallGroup(32,50)	50		1	12	32	32	32	32
Elementary abelian group:E32	51		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	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	?	sorting not done: all IDs 3, 4, 8, 12			Yes	No
1,3,20,8,0,0	1,4,24,32,32,32	4	?	sorting not done: all IDs 10, 13, 14, 41			No	No
1,3,28,0,0,0	1,4,32,32,32,32	3	?	sorting not done: all IDs 26, 32, 35			No	Yes
1,7,8,16,0,0	1,8,16,32,32,32	4	?	sorting not done: all IDs 5, 36, 37, 38			Yes	Yes
1,7,16,8,0,0	1,8,24,32,32,32	2	?	sorting not done: all IDs 11, 44			No	Yes
1,7,24,0,0,0	1,8,32,32,32,32	7	?	sorting not done: all IDs 2, 21, 23, 24, 29, 33, 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	?	sorting not done: all IDs 9, 40, 42			No	No
1,11,20,0,0,0	1,12,32,32,32,32	5	?	sorting not done: all IDs 6, 25, 30, 31, 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	?	sorting not done: all IDs 22, 28, 45, 48			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	?	sorting not done: all IDs 27, 34, 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 ] ]