#496084

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,29,24,26,39,43,22,28,37,45,20,21,27,38)(19,30,34,46,31,36,41,48,33,25,40,42,35,32,44,47);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,33)(18,25,28,30)(20,35,24,31)(21,36,26,32)(23,34,37,40)(27,44,39,41)(29,42,45,46)(38,48,43,47);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20,22,24)(18,26,28,21)(19,31,33,35)(23,27,37,39)(25,36,30,32)(29,43,45,38)(34,41,40,44)(42,48,46,47);(1,4,5,3)(2,8,6,7)(17,21,37,43,24,18,27,45,22,26,23,38,20,28,39,29)(19,32,40,48,31,30,44,42,33,36,34,47,35,25,41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=eg,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=di,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403280);