C₂×#14874 #607853

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

18878 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)(7,8,9,10)(12,14);(7,8)(9,10)(15,16,18,20)(17,21,22,19);(11,12)(13,14)(15,17,18,22)(16,19,20,21);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515049);