C₂×#25414 #6444211

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

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

3798 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);(7,8,10,9)(12,15)(13,16)(17,18);(3,4)(5,6)(7,9,10,8)(11,12,16,18,14,17,13,15);(4,6)(8,9);(11,12,16,18,14,17,13,15)

maximal subgroup of	C₂×#6445075, C₂×D₈×#121, C₂×#6578524, …
maximal normal subgroup of	C₂×#6445075, C₂×D₈×#121, C₂×#6578524, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351407);