#6407739

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,e^b=ei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6314935);