# 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 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]

### 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 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 $n$ $7^n$ total number of groups class 0 class 1 class 2 class 3 class 4 class 5 class 6