Groups of order 243

Statistics at a glance
Since $$243 = 3^5$$ is a prime power, and prime power order implies nilpotent, all groups of order 243 are nilpotent groups.

Summary information
Here, the rows are arithmetic functions that take values between $$0$$ and $$5$$, 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 $$67$$, which is the total number of groups of order $$243$$

Here is the GAP code to generate this information:

We use the function SortArithmeticFunctionSizes and some in-built and easy-to-code functions. The in-built functions used are Exponent and Logarithm (for [prime-base logarithm of exponent]]), NilpotencyClassOfGroup (for nilpotency class), DerivedLength (for derived length), and Rank (for minimum size of generating set). The functions FrattiniLength (for Frattini length), RankAsPGroup (for rank of a p-group), NormalRank (for normal rank of a p-group)), and CharacteristicRank (for characteristic rank of a p-group) are not in-built but are easily coded (follow links to get code).

gap> SortArithmeticFunctionSizes(3,5,G -> Log(Exponent(G),3)); [ [ 0, 0 ], [ 1, 4 ], [ 2, 49 ], [ 3, 11 ], [ 4, 2 ], [ 5, 1 ] ] gap> SortArithmeticFunctionSizes(3,5,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 46 ], [ 3, 17 ], [ 4, 2 ], [ 5, 1 ] ] gap> SortArithmeticFunctionSizes(3,5,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 7 ], [ 2, 28 ], [ 3, 26 ], [ 4, 6 ], [ 5, 0 ] ] gap> SortArithmeticFunctionSizes(3,5,DerivedLength); [ [ 0, 0 ], [ 1, 7 ], [ 2, 60 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ] ] gap> SortArithmeticFunctionSizes(3,5,Rank); [ [ 0, 0 ], [ 1, 1 ], [ 2, 29 ], [ 3, 30 ], [ 4, 6 ], [ 5, 1 ] ] gap> SortArithmeticFunctionSizes(3,5,RankAsPGroup); [ [ 0, 0 ], [ 1, 1 ], [ 2, 15 ], [ 3, 42 ], [ 4, 8 ], [ 5, 1 ] ] gap> SortArithmeticFunctionSizes(3,5,NormalRank); [ [ 0, 0 ], [ 1, 1 ], [ 2, 15 ], [ 3, 42 ], [ 4, 8 ], [ 5, 1 ] ] gap> SortArithmeticFunctionSizes(3,5,CharacteristicRank); [ [ 0, 0 ], [ 1, 3 ], [ 2, 17 ], [ 3, 39 ], [ 4, 7 ], [ 5, 1 ] ]