C₂×#62421 #10579174

order	512
nilpotency class	3
isoclinism	isoclinic class of #62253
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

8326 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);(4,7)(5,9)(12,15)(13,17)(16,18);(3,4)(5,8)(6,7)(9,10)(12,16)(15,18);(11,12,13,15,14,16,17,18);(4,7)(8,10);(3,5,6,9)(4,8,7,10)

maximal subgroup of	C₂×#10583262, C₂×#10583218, C₂×#10583287, …
maximal normal subgroup of	C₂×#10583262, C₂×#10583218, C₂×#10583287, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10486370);