# Groups of order 729

## Contents

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

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 $729 = 3^6$ 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 $0$ and $6$, 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 $504$, 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 ] ]```