C₂³×#398 #7627692

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

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

16586 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534888);