C₂×#25364 #6444131

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3982 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)(7,8,10,9);(3,4,6,5)(7,9,10,8)(11,12)(13,15)(14,17)(16,18);(8,9)(12,15)(13,18)(16,17);(3,5,6,4)

maximal subgroup of	C₂×#7521690, C₂×#7515405, C₂×#6444784, …
maximal normal subgroup of	C₂×#7521690, C₂×#7515405, C₂×#6444784, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351327);