C₂×#35518 #7628896

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
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

12382 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536092);