Element structure of groups of order 2^n

This article describes the element structure of groups of order 2^n, i.e., groups whose order is a power of 2.

$n$	$2^{n}$	Number of groups of order $2^{n}$	Information on groups	Information on element structure
0	1	1	trivial group	--
1	2	1	cyclic group:Z2	element structure of cyclic group:Z2
2	4	2	groups of order 4	element structure of groups of order 4
3	8	5	groups of order 8	element structure of groups of order 8
4	16	14	groups of order 16	element structure of groups of order 16
5	32	51	groups of order 32	element structure of groups of order 32
6	64	267	groups of order 64	element structure of groups of order 64
7	128	2328	groups of order 128	element structure of groups of order 128

Conjugacy class sizes

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

Number of size 1 conjugacy classes	Number of size 2 conjugacy classes	Number of size 4 conjugacy classes	Number of size 8 conjugacy classes	Total number of conjugacy classes	Order of group	Number of groups with these conjugacy class sizes	Description of the groups
1	0	0	0	1	1	1	trivial group only
2	0	0	0	2	2	1	cyclic group:Z2 only
4	0	0	0	4	4	2	cyclic group:Z4 and Klein four-group
8	0	0	0	8	8	3	abelian groups of order 8: cyclic group:Z8, direct product of Z4 and Z2, elementary abelian group:E8
2	3	0	0	5	8	2	the non-abelian groups of order 8: dihedral group:D8 and quaternion group
16	0	0	0	16	16	5	The abelian groups of order 16: cyclic group:Z16, direct product of Z4 and Z4, direct product of Z4 and Z4, direct product of Z8 and Z2, direct product of Z4 and V4, elementary abelian group:E16
4	6	0	0	10	16	6	The groups of order 16, class exactly two, the Hall-Senior family $\Gamma _{2}$: SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, direct product of D8 and Z2, direct product of Q8 and Z2, central product of D8 and Z4
2	3	2	0	7	16	3	The groups of order 16, class exactly three, the Hall-Senior family $\Gamma _{3}$: dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16
32	0	0	0	32	32	7	All the abelian groups of order 32
8	12	0	0	20	32	15	The Hall-Senior family (up to isocliny) $\Gamma _{2}$
2	15	0	0	17	32	2	The extraspecial groups (Hall-Senior family $\Gamma _{5}$): inner holomorph of D8 and central product of D8 and Q8
4	6	4	0	14	32	19	The Hall-Senior families $\Gamma _{3}$ (ten groups, class three) and $\Gamma _{4}$ (nine groups, class two)
2	3	6	0	11	32	5	The Hall-Senior families $\Gamma _{6}$ and $\Gamma _{7}$
2	7	0	2	11	32	3	The maximal class groups (family $\Gamma _{8}$): dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32

1-isomorphism

We say that two groups are 1-isomorphic groups if there exists a bijection between them that restricts to an isomorphism on all cyclic subgroups on both sides, i.e., there exists a 1-isomorphism between them. Below, we briefly describe the equivalence classes of groups of order $2^{n}$ up to 1-isomorphism:

$n$	Order $2^{n}$	Total number of groups (up to isomorphism)	Number of equivalence classes (up to 1-isomorphism)	List of equivalence classes of size more than one	More information
0	1	1	1	--	--
1	2	1	1	--	--
2	4	2	2	--	--
3	8	5	5	--	--
4	16	14	12	(direct product of Z8 and Z2, M16) and (direct product of Z4 and V4, central product of D8 and Z4)	Element structure of groups of order 16#1-isomorphism
5	32	51	38	(direct product of Z8 and Z4, semidirect product of Z8 and Z4 of M-type), (direct product of Z16 and Z2, M32), (direct product of Z4 and Z4 and Z2, SmallGroup(32,2), SmallGroup(32,24), SmallGroup(32,33)), (direct product of Z8 and V4, direct product of M16 and Z2, central product of D8 and Z8), (direct product of E8 and Z4, direct product of SmallGroup(16,13) and Z2), (direct product of Q8 and Z4, SmallGroup(32,32)), (direct product of SD16 and Z2, central product of D16 and Z4), (direct product of D8 and Z4, SmallGroup(32,30), SmallGroup(32,31)), (SmallGroup(32,27), generalized dihedral group for direct product of Z4 and Z4)