C₂×#25384 #6444331

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

3810 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)(11,12,14,18,15,13,16,17);(3,4,6,5)(7,8)(9,10)(11,13,14,17,15,12,16,18);(8,10)(12,17)(13,18)(14,16);(3,5,6,4)

maximal subgroup of	C₂×#7522087, C₂×#7519633, C₂×#6513518, …
maximal normal subgroup of	C₂×#7522087, C₂×#7519633, C₂×#6513518, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351527);