C₂×#18142 #6346384

order	512
nilpotency class	3
isoclinism	isoclinic class of #18132
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

5098 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(3,4)(5,7)(6,8)(9,10)(15,18);(4,7)(5,10)(8,9)(11,12)(13,15)(14,16)(17,18);(11,13,14,17)(12,15,16,18);(4,8)(7,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253580);