C₂×#25754 #6443136

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

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

5754 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 18 points:

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

maximal subgroup of	C₂×#6443175, C₂×#6573759, C₂×#6573784
maximal normal subgroup of	C₂×#6443175, C₂×#6573759, C₂×#6573784

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350332);