C₂×#39340 #7636713

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

11926 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 18 points:

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

maximal subgroup of	C₂×#7692992, C₂×#7671479, C₂×#7692342, …
maximal normal subgroup of	C₂×#7692992, C₂×#7671479, C₂×#7692342, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7543909);