C₂×#17068 #612275

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

9126 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519471);