C₂×#24439 #6437808

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 ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5250 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);(3,4,5,6)(12,17)(13,15)(14,18);(7,8)(9,10)(11,12,16,14)(13,17,15,18);(4,6)(8,10)(11,13,16,15)(12,17,14,18);(11,14,16,12)(13,18,15,17)

maximal subgroup of	C₂×#7519100, C₂×#6439861, C₂×#7516118, …
maximal normal subgroup of	C₂×#7519100, C₂×#6439861, C₂×#7516118, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345004);