C₂×#34743 #7627907

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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

12902 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);(7,8,10,9)(11,12)(13,16)(14,15)(17,18);(3,4)(5,6)(8,9);(4,6)(7,8,10,9)(12,15)(16,18);(4,6)(7,9,10,8);(11,13,14,17)(12,16,15,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535103);