C₂×#61569 #10577167

order	512
nilpotency class	3
isoclinism	isoclinic class of #61569
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 ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

9532 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10484363);