C₂×#60803 #10575506

order	512
nilpotency class	3
isoclinism	isoclinic class of #60750
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

9200 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)(5,9)(6,8)(7,10)(12,16)(13,14)(17,18);(3,5,6,10)(4,7,8,9);(5,10)(7,9)(11,12)(13,16)(14,18)(15,17);(4,8)(7,9);(11,13,15,14)(12,16,17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10482702);