Element structure of groups of order 32

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	Element structure page
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			2
SmallGroup(32,5)	5	20	${\mathrm{\Gamma }}_{2}j$	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			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	${\mathrm{\Gamma }}_{3}e$	3
Direct product of Z16 and Z2	16	6		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		1
Direct product of M16 and Z2	37	13	${\mathrm{\Gamma }}_{2}d$	2
SmallGroup(32,38)	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
SmallGroup(32,41)	41	25	${\mathrm{\Gamma }}_3{a}_3$	3
SmallGroup(32,42)	42			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		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
SmallGroup(32,50)	50	43	${\mathrm{\Gamma }}_5{a}_2$	2
Elementary abelian group:E32	51	1	$(1^5)$	1

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, as the table below shows:

Non-abelian member of pair	GAP ID	Abelian member of pair	GAP ID	Explanation for the 1-isomorphism	Description of the 1-isomorphism	Best perspective 1	Best perspective 2	Alternative perspective
M32	17	direct product of Z16 and Z2	16	The abelian group arises as the additive group of the generalized Baer cyclicity-preserving Lie ring of the non-abelian group for the extension with central subgroup cyclic group:Z8 and quotient Klein four-group	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	direct product of Z8 and Z4	3	The abelian group arises as the additive group of the generalized Baer cyclicity-preserving Lie ring of the non-abelian group for the extension with central subgroup cyclic group:Z4 and quotient group direct product of Z4 and Z2	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	direct product of Z8 and V4	36	The abelian group arises as the additive group of the generalized Baer cyclicity-preserving Lie ring of the non-abelian group for the extension with central subgroup direct product of Z4 and Z2 and quotient Klein four-group	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 and 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	direct product of Z8 and V4	36	The abelian group arises as the additive group of the generalized Baer cyclicity-preserving Lie ring of the non-abelian group for the extension with central subgroup cyclic group:Z4 and quotient elementary abelian group:E8	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	direct product of Z4 and Z4 and Z2	21	The abelian group arises as the additive group of the generalized Baer cyclicity-preserving Lie ring of the non-abelian group for the extension with central subgroup cyclic group:Z4 and quotient direct product of Z4 and Z2	generalized Baer correspondence between SmallGroup(32,24) and direct product of Z4 and Z4 and Z2	second cohomology group for trivial group action of direct product 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	direct product of E8 and Z4	45	The abelian group arises as the additive group of the generalized Baer cyclicity-preserving Lie ring of the non-abelian group for the extension with central subgroup direct product of Z4 and Z2 and quotient Klein four-group	generalized Baer correspondence between direct product of SmallGroup(16,13) and Z2 and direct product of E8 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 E8 and V4#Generalized Baer Lie rings	?
SmallGroup(32,2)	2	direct product of Z4 and Z4 and Z2	21	The abelian group arises as the additive group of the generalized Baer cyclicity-preserving Lie ring of the non-abelian group for the extension with central subgroup cyclic group:Z2 and quotient group direct product of Z4 and Z4	generalized Baer correspondence between SmallGroup(#2,2) 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	direct product of Z4 and Z4 and Z2	21	mystery	?

Groupings 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	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	semidirect product of Z8 and Z4 of M-type	4	${\mathrm{\Gamma }}_{2}i$	19
direct product of Z16 and Z2	16	M32	17	${\mathrm{\Gamma }}_{2}k$	22
direct product of Z4 and Z4 and Z2	21	SmallGroup(32,2), SmallGroup(32,24), SmallGroup(32,33)	2, 24, 33	${\mathrm{\Gamma }}_{2}h,{\mathrm{\Gamma }}_{2}f,{\mathrm{\Gamma }}_{4}d$	18, 16, 41
direct product of Z8 and V4	36	direct product of M16 and Z2, central product of D8 and Z8	37, 38	${\mathrm{\Gamma }}_{2}d,{\mathrm{\Gamma }}_{2}g$	13, 17
direct product of E8 and Z4	45	direct product of SmallGroup(16,13) and Z2	48	${\mathrm{\Gamma }}_{2}b$	10

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	${\mathrm{\Gamma }}_2{e}_2,{\mathrm{\Gamma }}_4{c}_3$	15, 40
direct product of SD16 and Z2 and central product of D16 and Z4	40, 42	${\mathrm{\Gamma }}_3{a}_2,{\mathrm{\Gamma }}_{3}b$	24, 26
direct product of D8 and Z4, SmallGroup(32,30), SmallGroup(32,31)	25, 30, 31	${\mathrm{\Gamma }}_2{e}_1,{\mathrm{\Gamma }}_4{c}_1,{\mathrm{\Gamma }}_4{c}_2$	14, 38, 39
SmallGroup(32,27), generalized dihedral group for direct product of Z4 and Z4	27, 34	${\mathrm{\Gamma }}_4{a}_1,{\mathrm{\Gamma }}_4{a}_2$	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 ] ]