C₂×#25451 #6446066

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2414 conjugacy classes of subgroups.

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

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

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

maximal subgroup of	C₂×#7520904, C₂×#7520857, C₂×#7519633, …
maximal normal subgroup of	C₂×#7520904, C₂×#7520857, C₂×#7519633, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353262);