C₂×#24239 #6442625

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
prime radical	C₂×#2537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5094 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)(9,10)(11,12,15,13)(14,16,18,17);(3,4,6,5)(11,13,15,12)(14,17,18,16);(8,10)(12,16)(13,17)(14,18);(3,5,6,4)

maximal subgroup of	C₂×D₄×#448
maximal normal subgroup of	C₂×D₄×#448

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349821);