Groups of order 5^n

Number of groups of small orders


Arithmetic functions
In the tables here, a row value of $$n$$ means we are looking at the groups of order $$5^n$$. The entry in a cell is the number of isomorphism classes of groups of order $$5^n$$ for which the function takes the value indicated in the column. Note that, for greater visual clarity, all zeros that occur after the last nonzero entry in a row are omitted and the corresponding entry is left blank.

Nilpotency class


Here is the GAP code to generate this information:

We use the function SortArithmeticFunctionSizes, which is not in-built but is easy to code (follow link to get code). We also use the in-built function NilpotencyClassOfGroup. Using these, we get:

gap> SortArithmeticFunctionSizes(5,0,NilpotencyClassOfGroup); [ [ 0, 1 ] ] gap> SortArithmeticFunctionSizes(5,1,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 1 ] ] gap> SortArithmeticFunctionSizes(5,2,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ] gap> SortArithmeticFunctionSizes(5,3,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ] gap> SortArithmeticFunctionSizes(5,4,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 5 ], [ 2, 6 ], [ 3, 4 ], [ 4, 0 ] ] gap> SortArithmeticFunctionSizes(5,5,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 7 ], [ 2, 30 ], [ 3, 31 ], [ 4, 9 ], [ 5, 0 ] ] gap> SortArithmeticFunctionSizes(5,6,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 11 ], [ 2, 149 ], [ 3, 386 ], [ 4, 99 ], [ 5, 39 ], [ 6, 0 ] ] gap> SortArithmeticFunctionSizes(5,7,NilpotencyClassOfGroup); [ [ 0, 0 ], [ 1, 15 ], [ 2, 7069 ], [ 3, 22652 ], [ 4, 3274 ], [ 5, 1188 ], [ 6, 99 ], [ 7, 0 ] ]

Note that for the $$5^7$$ case, the new version of the SmallGroup library (obtain here) is needed, and additional memory may need to be allocated using the command line option `-o' (e.g., gap -o 5G).

Here is the same information, now given in terms of the fraction of groups of a given order that are of a given nilpotency class. For ease of comparison, all fractions are written as decimals, rounded to the fourth decimal place.



Derived length


Here is the GAP code to generate this information:

We use the function SortArithmeticFunctionSizes, which is not in-built but is easy to code (follow link to get code). We also use the in-built function DerivedLength. Using these, we get:

gap> SortArithmeticFunctionSizes(5,0,NilpotencyClassOfGroup); [ [ 0, 1 ] ] gap> SortArithmeticFunctionSizes(5,0,DerivedLength); [ [ 0, 1 ] ] gap> SortArithmeticFunctionSizes(5,1,DerivedLength); [ [ 0, 0 ], [ 1, 1 ] ] gap> SortArithmeticFunctionSizes(5,2,DerivedLength); [ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ] gap> SortArithmeticFunctionSizes(5,3,DerivedLength); [ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ] gap> SortArithmeticFunctionSizes(5,4,DerivedLength); [ [ 0, 0 ], [ 1, 5 ], [ 2, 10 ], [ 3, 0 ], [ 4, 0 ] ] gap> SortArithmeticFunctionSizes(5,5,DerivedLength); [ [ 0, 0 ], [ 1, 7 ], [ 2, 70 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ] ] gap> SortArithmeticFunctionSizes(5,6,DerivedLength); [ [ 0, 0 ], [ 1, 11 ], [ 2, 657 ], [ 3, 16 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ] ] gap> SortArithmeticFunctionSizes(5,7,DerivedLength); [ [ 0, 0 ], [ 1, 15 ], [ 2, 33427 ], [ 3, 855 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ], [ 7, 0 ] ]

Note that for the $$5^7$$ case, the new version of the SmallGroup library (obtain here) is needed, and additional memory may need to be allocated using the command line option `-o' (e.g., gap -o 5G).

Here is the same information, now given in terms of the fraction of groups of a given order that are of a given derived length. For ease of comparison, all fractions are written as decimals, rounded to the fourth decimal place.



Frattini length


Here is the GAP code to generate this information:

We use the functions SortArithmeticFunctionSizes and FrattiniLength, which is not in-built but is easy to code (follow link to get code). Using these, we get:

gap> SortArithmeticFunctionSizes(5,0,FrattiniLength); [ [ 0, 1 ] ] gap> SortArithmeticFunctionSizes(5,1,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ] ] gap> SortArithmeticFunctionSizes(5,2,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 1 ] ] gap> SortArithmeticFunctionSizes(5,3,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 3 ], [ 3, 1 ] ] gap> SortArithmeticFunctionSizes(5,4,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 11 ], [ 3, 2 ], [ 4, 1 ] ] gap> SortArithmeticFunctionSizes(5,5,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 62 ], [ 3, 11 ], [ 4, 2 ], [ 5, 1 ] ] gap> SortArithmeticFunctionSizes(5,6,FrattiniLength); [ [ 0, 0 ], [ 1, 1 ], [ 2, 546 ], [ 3, 122 ], [ 4, 12 ], [ 5, 2 ], [ 6, 1 ] ] 