C₂×#24680 #6449806

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

2434 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);(3,4,5,6)(7,8,10,11)(9,13,14,12);(8,11)(12,13)(16,19)(17,21)(20,22);(4,6)(15,16,18,20)(17,22,21,19);(7,9,10,14)(8,12,11,13)

maximal subgroup of	C₂×#7522705, C₂×#7519801, C₂×#6481774, …
maximal normal subgroup of	C₂×#7522705, C₂×#7519801, C₂×#6481774, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357002);