# 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 centralizerBounding facts: size of conjugacy class is bounded by order of derived subgroupCounting 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 | (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 | |

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 | (class three, ten groups numbers 23--32) and (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 | (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 | and |

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 |

### 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 | (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 | |

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 | (class three, ten groups numbers 23--32) and (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 | (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 | and |

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 |

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 , with a prime and . There are, however, counterexamples for .

- 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] |
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
8 | 1 | This is SmallGroup(32,33). |

Here are the details:

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

### 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 | 19 | |

direct product of Z16 and Z2 | 16 | 1 | M32 | 17 | 22 | |

direct product of Z4 and Z4 and Z2 | 21 | 3 | SmallGroup(32,2), SmallGroup(32,24), SmallGroup(32,33) | 2, 24, 33 | 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 | 13, 17 | |

direct product of E8 and Z4 | 45 | 1 | direct product of SmallGroup(16,13) and Z2 | 48 | 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 | 15, 40 | |

direct product of SD16 and Z2 and central product of D16 and Z4 | 40, 42 | 24, 26 | |

direct product of D8 and Z4, SmallGroup(32,30), SmallGroup(32,31) | 25, 30, 31 | 14, 38, 39 | |

SmallGroup(32,27), generalized dihedral group for direct product of Z4 and Z4 | 27, 34 | 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 or is odd, while all the other numbers are even. The total number of roots is even for all .

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 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 |

### 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 |