C₂×#15380 #608913

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6158 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(4,6)(7,8,10,9)(16,19)(17,21);(8,9)(12,14);(7,8,10,9)(11,12)(13,14)(15,16)(17,20)(18,19)(21,22);(3,4)(5,6)(7,9,10,8)(15,17,18,21)(16,20,19,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 516109);