# Groups of order 243

From Groupprops

This article gives information about, and links to more details on, groups of order 243

See pages on algebraic structures of order 243| See pages on groups of a particular order

## Statistics at a glance

Since is a prime power, and prime power order implies nilpotent, all groups of order 243 are nilpotent groups.

Quantity | Value | Explanation |
---|---|---|

Total number of groups up to isomorphism | 67 | |

Number of abelian groups | 7 | Equals the number of unordered integer partitions of 5, see classification of finite abelian groups |

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

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

Number of groups of nilpotency class exactly four (i.e., maximal class groups) |
6 |

## Arithmetic functions

### Summary information

Here, the rows are arithmetic functions that take values between and , and the columns give the possible values of these functions. The entry in each cell is the number of isomorphism classes of groups for which the row arithmetic function takes the column value. Note that all the row value sums must equal , which is the total number of groups of order

Arithmetic function | Value 0 | Value 1 | Value 2 | Value 3 | Value 4 | Value 5 |
---|---|---|---|---|---|---|

prime-base logarithm of exponent | 0 | 4 | 49 | 11 | 2 | 1 |

Frattini length | 0 | 1 | 46 | 17 | 2 | 1 |

nilpotency class | 0 | 7 | 28 | 26 | 6 | 0 |

derived length | 0 | 7 | 60 | 0 | 0 | 0 |

minimum size of generating set | 0 | 1 | 29 | 30 | 6 | 1 |

rank as p-group | 0 | 1 | 15 | 42 | 8 | 1 |

normal rank as p-group | 0 | 1 | 15 | 42 | 8 | 1 |

characteristic rank as p-group | 0 | 1 | 17 | 39 | 7 | 1 |