# Groups of prime-fifth order

From Groupprops

This article is about the groups of prime-fifth order, i.e., order where is an odd prime. The cases (see groups of order 32) and (see groups of order 243) are somewhat different from the general case .

is the smallest prime power for which the number of groups of that order is not eventually constant, but rather, is given by a nonconstant PORC function in keeping with Higman's PORC conjecture.

## Statistics at a glance

Quantity | Value case | Value case | PORC function for |
---|---|---|---|

Total number of groups | 51 | 67 | |

Number of abelian groups | 7 | 7 | 7 |

Number of groups of nilpotency class exactly two |
26 | 28 | |

Number of groups of nilpotency class exactly three |
15 | 26 | |

Number of groups of nilpotency class exactly four (maximal class groups) |
3 | 6 |

## Particular cases

Prime number | Number of groups of order | Information on groups of order | |
---|---|---|---|

2 | 32 | 51 | groups of order 32 -- somewhat anomalous |

3 | 243 | 67 | groups of order 243 -- somewhat anomalous |

5 | 3125 | 77 | groups of order 3125 |

7 | 16807 | 83 | groups of order 16807 |

11 | 161051 | 87 | groups of order 161051 |