# Groups of prime-sixth order

From Groupprops

This article is about the groups of prime-sixth order, i.e., order where is a prime number. The cases (see groups of order 64) and (see groups of order 729) are somewhat different from the general case .

The number of groups of order for is not a constant. Rather, it is given by a non-constant 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 | 267 | 504 | |

Number of abelian groups | 11 | 11 | 11 |

Number of groups of nilpotency class exactly two |
117 | 133 | |

Number of groups of nilpotency class exactly three |
114 | 282 | ? |

Number of groups of nilpotency class exactly four |
22 | 71 | ? |

Number of groups of nilpotency class exactly five |
3 | 7 | ? |

## Particular cases

Note that the number of groups of order is given by the PORC function for .

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

2 | 64 | 267 | groups of order 64 -- somewhat anomalous |

3 | 729 | 504 | groups of order 729 -- somewhat anomalous |

5 | 15625 | 684 | groups of order 15625 |

7 | 117649 | 860 | groups of order 117649 |

11 | 1771561 | 1192 | groups of order 1771561 |