C₂×#28809 #6465736

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2370 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

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

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

maximal subgroup of	C₂×#6583220, C₂×#6465861
maximal normal subgroup of	C₂×#6583220, C₂×#6465861

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372932);