C₂×#20054 #6359380

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×D₄×D₈	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2196 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)(5,10)(8,9)(11,12)(13,16)(14,17)(15,18);(3,4,6,9)(5,8,10,7)(11,13,15,14)(12,16,18,17);(12,17)(13,14)(16,18);(4,8)(5,10)(7,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266576);