C₂×#19671 #6357243

order	512
nilpotency class	3
isoclinism	isoclinic class of #19671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2738 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);(4,7)(5,9)(8,10)(12,15)(13,17)(16,18);(3,4,5,7,6,8,9,10)(12,16)(15,18);(11,12,13,15,14,16,17,18);(4,8)(7,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264439);