#495442

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄

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

490 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,27,23,28,37,44,21,26,29,38,36,43,39,48)(19,24,31,33,25,41,42,47,30,40,45,20,32,34,46,35);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,29,45,23,25,39,46,21,30,22,31,36,32,37,42)(18,24,38,20,28,41,48,35,26,40,27,33,43,34,44,47);(9,12,13,11)(10,14,15,16)(17,20,21,33)(18,25,26,32)(19,28,30,43)(22,34,29,41)(23,35,36,47)(24,39,40,37)(27,42,38,46)(31,44,45,48);(1,3,4,7)(2,6,5,8)(19,30)(20,33)(24,40)(25,32)(31,45)(34,41)(35,47)(42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,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, 402638);