C₂×#59051 #7666337

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

12602 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)(11,12,14,16)(13,17,18,15);(3,4,6,5)(8,10)(13,18)(15,17);(11,13,14,18)(12,15,16,17);(7,8)(9,10)(11,12)(13,15)(14,16)(17,18);(3,5,6,4)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7573533);