# 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.

## Contents

## Facts known for general primes

## Facts known for specific primes

### The case

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.

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 |

### The case

### The case

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.

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

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 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.

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 |