C₂×#18529 #6348074

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×D₄²	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

6046 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255270);