C₂×#61959 #10578749

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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

8066 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)(13,18)(14,16)(15,17);(4,7)(8,10);(3,5,6,9)(4,8,7,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10485945);