C₂×#60916 #10575614

order	512
nilpotency class	3
isoclinism	isoclinic class of #60750
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 ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

8052 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)(11,12)(13,15)(14,16)(17,18);(3,4)(5,8)(6,7)(9,10);(4,7)(8,10)(12,15)(13,18)(16,17);(3,5,6,9)(4,8,7,10);(12,16)(15,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10482810);