# Groups of order 729

This article gives information about, and links to more details on, groups of order 729

See pages on algebraic structures of order 729| See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 729. See also more detailed information on specific subtopics through the links:

Information type | Page summarizing information for groups of order 729 |
---|---|

element structure (element orders, conjugacy classes, etc.) | element structure of groups of order 729 |

subgroup structure | subgroup structure of groups of order 729 |

linear representation theory | linear representation theory of groups of order 729 projective representation theory of groups of order 729 modular representation theory of groups of order 729 |

endomorphism structure, automorphism structure | endomorphism structure of groups of order 729 |

group cohomology | group cohomology of groups of order 729 |

## Statistics at a glance

To understand these in a broader context, see

groups of order 3^n|groups of prime-sixth order

Since is a prime power and prime power order implies nilpotent, all groups of order 729 are nilpotent groups.

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

Total number of groups up to isomorphism | 504 | |

Number of abelian groups | 11 | Equals the number of unordered integer partitions of 6. See classification of finite abelian groups. |

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

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

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

Number of groups of nilpotency class exactly five (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 number of groups of order 729.

Arithmetic function | Value 0 | Value 1 | Value 2 | Value 3 | Value 4 | Value 5 | Value 6 |
---|---|---|---|---|---|---|---|

prime-base logarithm of exponent | 0 | 8 | 401 | 80 | 12 | 2 | 1 |

Frattini length | 0 | 1 | 355 | 133 | 12 | 2 | 1 |

nilpotency class | 0 | 11 | 133 | 282 | 71 | 6 | 0 |

derived length | 0 | 11 | 493 | 0 | 0 | 0 | 0 |

minimum size of generating set | 0 | 1 | 100 | 313 | 82 | 7 | 1 |

Here is the GAP code to generate this information: <toggledisplay>

We use the function SortArithmeticFunctionSizes (not in-built, needs to be coded, follow the link to get the code) as well as various in-built and coded GAP functions. The in-built functions are: Exponent and Logarithm (for prime-base logarithm of exponent), GAP:NilpotencyClassOfGroup (for nilpotency class), DerivedLength (for derived length), Rank (for minimum size of generating set). The coded functions include FrattiniLength (for Frattini length).

gap> SortArithmeticFunctionSizes(3,6,G -> Log(Exponent(G),3)); [ [ 0, 0 ], [ 1, 8 ], [ 2, 401 ], [ 3, 80 ], [ 4, 12 ], [ 5, 2 ], [ 6, 1 ] ] gap> SortArithmeticFunctionSizes(3,6,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 355 ], [ 3, 133 ], [ 4, 12 ], [ 5, 2 ], [ 6, 1 ] ] gap> SortArithmeticFunctionSizes(3,6,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 11 ], [ 2, 133 ], [ 3, 282 ], [ 4, 71 ], [ 5, 7 ], [ 6, 0 ] ] gap> SortArithmeticFunctionSizes(3,6,DerivedLength); [ [ 0, 0 ], [ 1, 11 ], [ 2, 493 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ] ] gap> SortArithmeticFunctionSizes(3,6,Rank); [ [ 0, 0 ], [ 1, 1 ], [ 2, 100 ], [ 3, 313 ], [ 4, 82 ], [ 5, 7 ], [ 6, 1 ] ]