Groups of order 2^n

Number of groups of small orders

Exponent $n$	Value $2^{n}$	Number of groups of order $2^{n}$	Reason/Explanation/List
1	2	1	Only cyclic group:Z2; see equivalence of definitions of group of prime order
2	4	2	Only cyclic group:Z4 and Klein four-group; see also groups of order 4 and classification of groups of prime-square order
3	8	5	See groups of order 8; see classification of groups of prime-cube order
4	16	14	See groups of order 16
5	32	51	See groups of order 32
6	64	267	See groups of order 64
7	128	2328	See groups of order 128
8	256	56092	See groups of order 256
9	512	10494213	See groups of order 512
10	1024	49487365422	See groups of order 1024

Arithmetic functions

In the tables here, a row value of $n$ means we are looking at the groups of order $2^{n}$ . The entry in a cell is the number of isomorphism classes of groups of order $2^{n}$ for which the function takes the value indicated in the column. Note that, for greater visual clarity, all zeros that occur after the last nonzero entry in a row are omitted and the corresponding entry is left blank.

Nilpotency class

$n$	$2^{n}$	class 0	class 1	class 2	class 3	class 4	class 5	class 6
0	1	1
1	2	0	1
2	4	0	2
3	8	0	3	2
4	16	0	5	6	3
5	32	0	7	26	15	3
6	64	0	11	117	114	22	3
7	128	0	15	947	1137	197	29	3

Derived length

$n$	$2^{n}$	length 0	length 1	length 2	length 3
0	1	1
1	2	0	1
2	4	0	2
3	8	0	3	2
4	16	0	5	9
5	32	0	7	44
6	64	0	11	256
7	128	0	15	2299	14

Frattini length

$n$	$2^{n}$	length 0	length 1	length 2	length 3	length 4	length 5	length 6	length 7
0	1	1
1	2	0	1
2	4	0	1	1
3	8	0	1	3	1
4	16	0	1	7	5	1
5	32	0	1	23	21	5	1
6	64	0	1	94	139	27	5	1
7	128	0	1	816	1276	202	27	5	1