C₂×#37652 #7635073

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

14174 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)(13,14)(15,16)(17,18);(4,6)(12,16)(14,17);(3,4)(5,6)(7,8)(9,10);(11,13)(12,17)(14,16)(15,18);(3,4)(5,6)(11,14,15,17)(12,13,16,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542269);