C₂×#39355 #7636635

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

13298 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);(8,10)(11,12)(13,17)(14,15)(16,18);(4,5)(12,15)(16,17);(3,4,6,5)(7,8,9,10);(11,13,14,18)(12,16,15,17);(3,5,6,4)(11,12)(13,16)(14,15)(17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7543831);