# Groups of prime power order

From Groupprops

This is a specific information page. For the definition of the term and related information, see group of prime power order

This article gives information about groups of prime power order, i.e., order where is a prime number and is a nonnegative integer. The case , i.e., groups of order 1, only includes the trivial group, so we exclude this case from the discussion below.

## Information

## Counts

Generic formula for number of groups of order (if it exists) for large enough | Minimum from which the formula becomes valid | Number of groups of order | Number of groups of order | Number of groups of order | Number of groups of order | Number of groups of order | |
---|---|---|---|---|---|---|---|

1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 |

2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

3 | 5 | 2 | 5 | 5 | 5 | 5 | 5 |

4 | 15 | 3 | 14 | 15 | 15 | 15 | 15 |

5 | 5 | 51 | 67 | 77 | 83 | 87 | |

6 | 5 | 267 | 504 | 684 | 860 | 1192 | |

7 | 7 | 2328 | 9310 | 34297 | 113147 | 750735 | |

8 | unknown | 56092 | 1396077 | unknown | unknown | unknown | |

9 | unknown | 10494213 | unknown | unknown | unknown | unknown | |

10 | unknown | 49487365422 | unknown | unknown | unknown | unknown |