Groups of order 729

Statistics at a glance
Since $$729 = 3^6$$ is a prime power and prime power order implies nilpotent, all groups of order 729 are nilpotent groups.

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.

Here is the GAP code to generate this information:

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 ] ]