Groups of order 64

This article gives information about, and links to more details on, groups of order 64
See pages on algebraic structures of order 64 | See pages on groups of a particular order

Statistics at a glance

To understand these in a broader context, see
groups of order 2^n|groups of prime-sixth order

Since $64=2^6$ is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.

Quantity	Value	Explanation
Number of groups up to isomorphism	267
Number of abelian groups up to isomorphism	11	Equals the number of unordered integer partitions of $6$. See also classification of finite abelian groups.
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, i.e., maximal class groups	3	The dihedral group, semidihedral group, and generalized quaternion group.

Arithmetic functions

Summary information

Here, the rows are arithmetic functions that take values between $0$ and $6$, 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 $267$, which is the total number of groups of order $64$.

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