Nilpotency class distribution of finite p-groups

This article gives both numerical information and links to known facts/conjectures about the distribution of nilpotency class among finite p-groups, i.e., how many p-groups there are of a given nilpotency class.

Facts known for specific primes

The case $p = 2$

$n$	$2^n$	total number of groups	class 0	class 1	class 2	class 3	class 4	class 5	class 6	class 7
0	1	1	1
1	2	1	0	1
2	4	2	0	2
3	8	5	0	3	2
4	16	14	0	5	6	3
5	32	51	0	7	26	15	3
6	64	267	0	11	117	114	22	3
7	128	2328	0	15	947	1137	197	29	3
8	256	56092	0	22	31742	21325	2642	320	38	3

Here is the GAP code to generate this information: [SHOW MORE]

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 functions, the above data can be generated as follows:

gap> SortArithmeticFunctionSizes(2,0,NilpotencyClassOfGroup);
[ [ 0, 1 ] ]
gap> SortArithmeticFunctionSizes(2,1,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(2,2,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(2,3,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(2,4,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 6 ], [ 3, 3 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(2,5,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 26 ], [ 3, 15 ], [ 4, 3 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(2,6,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 117 ], [ 3, 114 ], [ 4, 22 ], [ 5, 3 ], [ 6, 0 ] ]
gap> SortArithmeticFunctionSizes(2,7,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 15 ], [ 2, 947 ], [ 3, 1137 ], [ 4, 197 ], [ 5, 29 ], [ 6, 3 ], [ 7, 0 ] ]
gap> SortArithmeticFunctionSizes(2,8,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 22 ], [ 2, 31742 ], [ 3, 21325 ], [ 4, 2642 ], [ 5, 320 ], [ 6, 38 ], [ 7, 3 ], [ 8, 0 ] ]

Although the command can in principle be run for $2^8$, it requires a huge memory allocation (this can be controlled setting the "-o" command line option when starting GAP). To obtain the result for $2^8$, it is best to split the list of all small groups of order $2^8$ into sub-lists and run the function on each.

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.

$n$	$2^n$	total number of groups	average of values (equal weighting on all groups)	class 0	class 1	class 2	class 3	class 4	class 5	class 6	class 7
0	1	1	0	1
1	2	1	1	0	1
2	4	2	1	0	1
3	8	5	1.4	0	0.6000	0.4000
4	16	14	1.8571	0	0.3571	0.4286	0.2143
5	32	51	2.2745	0	0.1373	0.5098	0.2941	0.0588
6	64	267	2.5843	0	0.0412	0.4382	0.4270	0.0824	0.0112
7	128	2328	2.6937	0	0.0064	0.4068	0.4884	0.0846	0.01245	0.0013
8	256	56092	2.4941	0	0.0004	0.5659	0.3802	0.0471	0.0057	0.0007	0.0001

Below is the information for the probability distribution by nilpotency class using the cohomology tree probability distribution: [SHOW MORE]

$n$	$2^n$	total number of groups	weighted average nilpotency class under the distribution	class 0	class 1	class 2	class 3	class 4	class 5	class 6	class 7
0	1	1	0	1
1	2	1	1	0	1
2	4	2	1	0	1
3	8	5	1.25	0	0.7500	0.2500
4	16	14	1.6172	0	0.4766	0.4297	0.0938
5	32	51	1.9889	0	0.2772	0.4684	0.2427	0.0117
6	64	267	2.3329	0	0.1533	0.4231	0.3623	0.0598	0.0015

The case $p = 5$

$n$	$5^n$	total number of groups	class 0	class 1	class 2	class 3	class 4	class 5	class 6
0	1	1	1
1	5	1	0	1
2	25	2	0	2
3	125	5	0	3	2
4	625	15	0	5	6	4
5	3125	77	0	7	30	31	9
6	15625	684	0	11	149	386	99	39
7	78125	34297	0	15	7069	22652	3274	1188	99

Here is the GAP code to generate this information: [SHOW MORE]

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.

$n$	$5^n$	total number of groups	class 0	class 1	class 2	class 3	class 4	class 5	class 6
0	1	1	1
1	5	1	0	1
2	25	2	0	1
3	125	5	0	0.6000	0.4000
4	625	15	0	0.3333	0.4000	0.2667
5	3125	77	0	0.0909	0.3896	0.4026	0.1169
6	15625	684	0	0.0161	0.2178	0.5643	0.1447	0.5702
7	78125	34297	0	0.0044	0.2063	0.6605	0.0955	0.0346	0.0289

The case $p = 7$

$n$	$7^n$	total number of groups	class 0	class 1	class 2	class 3	class 4	class 5	class 6
0	1	1	1
1	7	1	0	1
2	49	2	0	2
3	343	5	0	3	2
4	2401	15	0	5	6	4
5	16807	83	0	7	32	33	11
6	117649	860	0	11	165	508	133	43

Here is the GAP code to generate this information: [SHOW MORE]

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 functions, the above data can be generated as follows:

gap> SortArithmeticFunctionSizes(7,0,NilpotencyClassOfGroup);
[ [ 0, 1 ] ]
gap> SortArithmeticFunctionSizes(7,1,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(7,2,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(7,3,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(7,4,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 6 ], [ 3, 4 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(7,5,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 32 ], [ 3, 33 ], [ 4, 11 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(7,6,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 165 ], [ 3, 508 ], [ 4, 133 ], [ 5, 43 ], [ 6, 0 ] ]

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.

$n$	$7^n$	total number of groups	class 0	class 1	class 2	class 3	class 4	class 5	class 6
0	1	1	1
1	7	1	0	1
2	49	2	0	1
3	343	5	0	0.6000	0.4000
4	2401	15	0	0.3333	0.4000	0.2667
5	16807	83	0	0.0843	0.3855	0.3976	0.1325
6	117649	860	0	0.0128	0.1919	0.5907	0.1547	0.0500