C₂×#25731 #6443083

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6270 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(4,5)(11,12,14,16)(13,18,17,15);(3,4,6,5)(7,8)(9,10)(12,16)(13,17);(8,10)(11,13,15,12,14,17,18,16);(3,5,6,4)(12,16)(13,17)

maximal subgroup of	C₂×#7519100, C₂×#7521687, C₂×#6454561, …
maximal normal subgroup of	C₂×#7519100, C₂×#7521687, C₂×#6454561, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350279);