C₂×#35358 #7628712

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
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

14154 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)(8,10);(7,8,9,10)(11,12)(13,17)(14,15)(16,18);(3,4)(5,6)(12,15)(16,17);(11,12)(13,16)(14,15)(17,18);(11,13)(12,16)(14,18)(15,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535908);