# Nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order

This article gives a fact that is true forsmallgroups of prime power order.More specifically, it is true for all groups of order where is at most equal to 4.

See more such facts| See more facts true for prime powers up to the same maximum power 4

## Contents

## Statement

Suppose is a prime number and . Then, for a group of order , the Nilpotency class (?) of the group determines its conjugacy class size statistics.

Here is the complete list:

(prime-base logarithm of order) | (nilpotency class) | number of conjugacy classes of size 1 | number of conjugacy classes of size | number of conjugacy classes of size |
---|---|---|---|---|

0 | 0 | 1 | 0 | 0 |

1 | 1 | 0 | 0 | |

2 | 1 | 0 | 0 | |

3 | 1 | 0 | 0 | |

3 | 2 | 0 | ||

4 | 1 | 0 | 0 | |

4 | 2 | 0 | ||

4 | 3 |