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

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	?	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	?	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(32,29) (ID:29)			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 ] ]