C₂×#59313 #7666499

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
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

11802 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);(4,5)(8,9)(11,12,15,17)(13,14,16,18);(3,4,6,5)(8,9)(11,13)(12,14)(15,16)(17,18);(7,8,10,9);(7,9,10,8)(11,14)(12,16)(13,17)(15,18);(3,5,6,4)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7573695);