Element structure of groups of order 32

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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]

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] 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] 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] 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] 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] 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] 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.

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

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] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] 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]

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]

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]