C₂×#20170 #6359520

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2211 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266716);