C₂×#59128 #7666369

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

11964 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);(8,9)(12,16)(14,17);(4,6)(8,9)(11,12)(13,17)(14,18)(15,16);(7,8,10,9)(11,13,15,18)(12,17,16,14);(3,4)(5,6)(7,9,10,8);(11,14,15,17)(12,18,16,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7573565);