C₂×#15172 #608431

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

direct product

C₂×#15172

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

10062 conjugacy classes of subgroups; only the 1282 normal subgroups are shown.

normal subgroups only

This group has 10062 conjugacy classes of subgroups — too many to draw. Only its 1282 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(4,6)(8,9)(12,13)(17,21)(20,22);(7,8,10,9)(16,19)(20,22);(11,12,14,13)(15,16)(17,20)(18,19)(21,22);(3,4,5,6)(7,9,10,8)(11,13,14,12)(15,17)(16,20)(18,21)(19,22)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72228208, #72228215, #72268873, …
maximal subgroup of	#72222661, C₂×#15172⋊C₃, #72222727, …
maximal normal subgroup of	#72222661, C₂×#15172⋊C₃, #72222727, …
maximal quotient of	C₂×#158420, C₂×#158361, C₂×#154647, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515627);