Groups of order 64
Statistics at a glance
| Quantity | Value |
|---|---|
| Number of groups up to isomorphism | 267 |
| Number of abelian groups up to isomorphism | 11 |
| Number of groups of class exactly two | 117 |
| Number of groups of class exactly three | 114 |
| Number of groups of class exactly four | 22 |
| Number of groups of class exactly five | 3 |
Arithmetic functions
Summary information
Here, the rows are arithmetic functions that take values between and , 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 , which is the total number of groups of order .
| Arithmetic function | Value 0 | Value 1 | Value 2 | Value 3 | Value 4 | Value 5 | Value 6 |
|---|---|---|---|---|---|---|---|
| prime-base logarithm of exponent | 0 | 1 | 96 | 137 | 27 | 5 | 1 |
| Frattini length | 0 | 1 | 94 | 139 | 27 | 5 | 1 |
| nilpotency class | 0 | 11 | 117 | 114 | 22 | 3 | 0 |
| derived length | 0 | 11 | 256 | 0 | 0 | 0 | 0 |
| minimum size of generating set | 0 | 1 | 53 | 137 | 68 | 7 | 1 |
| rank of a p-group | 0 | 2 | 54 | 150 | 55 | 5 | 1 |
| normal rank of a p-group | 0 | 4 | 87 | 122 | 48 | 5 | 1 |
| characteristic rank of a p-group |