C₂×#25401 #6444346

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3400 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)(11,12,16,13,14,17,15,18);(3,4,6,5)(7,8)(9,10)(11,13,16,17,14,18,15,12);(8,10)(12,13)(15,16)(17,18);(3,5,6,4)(12,17)(13,18)

maximal subgroup of	C₂×#6583368, C₂×#6456529, C₂×#7522311, …
maximal normal subgroup of	C₂×#6583368, C₂×#6456529, C₂×#7522311, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351542);