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