C₂×#52096 #7633455

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
socle	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

12580 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);(3,4)(5,6)(11,12)(13,14)(15,17)(16,18);(7,8)(9,10)(11,13)(12,16)(14,17)(15,18);(4,6)(8,10);(4,6)(12,17)(13,18);(11,14,15,16)(12,13,17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540651);