# Element structure of groups of order 64

This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 64.

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## 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

### 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 64 mod 3, and hence congruent to 1 mod 3.

Number of size 1 conjugacy classes | Number of size 2 conjugacy classes | Number of size 4 conjugacy classes | Number of size 8 conjugacy classes | Number of size 16 conjugacy classes | Total number of conjugacy classes | Total number of groups | Nilpotency class(es) attained by these groups | Hall-Senior family/families | List of GAP IDs (second part) |
---|---|---|---|---|---|---|---|---|---|

64 | 0 | 0 | 0 | 0 | 64 | 11 | 1 | , i.e., all the abelian groups of order 64 | [SHOW MORE] |

16 | 24 | 0 | 0 | 0 | 40 | 31 | 2 | [SHOW MORE] | |

8 | 12 | 8 | 0 | 0 | 28 | 60 | 2,3 | (class three) and (class two) | [SHOW MORE] |

8 | 0 | 14 | 0 | 0 | 22 | 10 | 2 | ? | [SHOW MORE] |

4 | 30 | 0 | 0 | 0 | 34 | 7 | 2 | [SHOW MORE] | |

4 | 14 | 0 | 4 | 0 | 22 | 23 | 3, 4 | [SHOW MORE] | |

4 | 12 | 9 | 0 | 0 | 25 | 15 | 2 | [SHOW MORE] | |

4 | 6 | 12 | 0 | 0 | 22 | 38 | 2,3 | and | [SHOW MORE] |

4 | 4 | 9 | 2 | 0 | 19 | 31 | 3 | [SHOW MORE] | |

4 | 2 | 6 | 4 | 0 | 16 | 9 | 3 | [SHOW MORE] | |

4 | 0 | 15 | 0 | 0 | 19 | 5 | 2 | [SHOW MORE] | |

2 | 15 | 0 | 0 | 2 | 19 | 3 | 5 | [SHOW MORE] | |

2 | 9 | 11 | 0 | 0 | 22 | 3 | 3 | [SHOW MORE] | |

2 | 5 | 5 | 4 | 0 | 16 | 9 | 3,4 | [SHOW MORE] | |

2 | 3 | 8 | 3 | 0 | 16 | 6 | 3 | [SHOW MORE] | |

2 | 1 | 5 | 5 | 0 | 13 | 6 | 4 | [SHOW MORE] |

### Grouping by cumulative conjugacy class sizes (number of elements)

Number of elements in (size at most 1) conjugacy classes | Number of elements in (size at most 2) conjugacy classes | Number of elements in (size at most 4) conjugacy classes | Number of elements in (size at most 8) conjugacy classes | Number of elements in (size at most 16) conjugacy classes | Total number of conjugacy classes | Total number of groups | Nilpotency class(es) attained by these groups | Hall-Senior family/families | List of GAP IDs (second part) |
---|---|---|---|---|---|---|---|---|---|

64 | 64 | 64 | 64 | 64 | 64 | 11 | 1 | , i.e., all the abelian groups of order 64 | [SHOW MORE] |

16 | 64 | 64 | 64 | 64 | 40 | 31 | 2 | [SHOW MORE] | |

8 | 32 | 64 | 64 | 64 | 28 | 60 | 2,3 | (class three) and (class two) | [SHOW MORE] |

8 | 8 | 64 | 64 | 64 | 22 | 10 | 2 | ? | [SHOW MORE] |

4 | 64 | 64 | 64 | 64 | 34 | 7 | 2 | [SHOW MORE] | |

4 | 32 | 32 | 64 | 64 | 22 | 23 | 3, 4 | [SHOW MORE] | |

4 | 28 | 64 | 64 | 64 | 25 | 15 | 2 | [SHOW MORE] | |

4 | 16 | 64 | 64 | 64 | 24 | 38 | 2,3 | and | [SHOW MORE] |

4 | 12 | 48 | 64 | 64 | 19 | 31 | 3 | [SHOW MORE] | |

4 | 8 | 32 | 64 | 64 | 16 | 9 | 3 | [SHOW MORE] | |

4 | 4 | 64 | 64 | 64 | 19 | 5 | 2 | [SHOW MORE] | |

2 | 32 | 32 | 32 | 64 | 19 | 3 | 5 | [SHOW MORE] | |

2 | 20 | 64 | 64 | 64 | 22 | 3 | 3 | [SHOW MORE] | |

2 | 12 | 32 | 64 | 64 | 16 | 9 | 3,4 | [SHOW MORE] | |

2 | 8 | 40 | 64 | 64 | 16 | 6 | 3 | [SHOW MORE] | |

2 | 4 | 24 | 64 | 64 | 13 | 6 | 4 | [SHOW MORE] |

Note that it is *not* true that the cumulative conjugacy class size statistics values divide the order of the group in all cases. There are a few counterexamples in the table above, as we can see values such as 12, 20, 28, and 40. is the smallest prime power where such examples exist. See also:

- 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

## 1-isomorphism

### Pairs where one of the groups is abelian

There are 29 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. Of these, the only example of a group that is not of nilpotency class two is SmallGroup(64,25) (GAP ID: 25):

Here is a summary version:

A total of 23 of the 29 1-isomorphisms are explained using the explanations here. Here is a long version:

### Grouping by abelian member

Of the 11 abelian groups of order 64, 9 are 1-isomorphic to non-abelian groups. The only two that aren't are cyclic group:Z64, on account of the fact that finite group having the same order statistics as a cyclic group is cyclic, and elementary abelian group:E64, on account of the fact that exponent two implies abelian.