#6840509

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1343 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)(4,8)(5,6)(9,10,13,15)(11,16,12,14)(17,18)(19,24)(20,25)(21,35)(22,27)(23,28)(26,36)(29,38)(30,39)(31,40)(32,41)(33,44)(34,42)(37,43)(45,47)(46,48);(9,10)(11,14)(12,16)(13,15)(17,19,23,30)(18,24,28,39)(20,31,32,33)(21,22,36,37)(25,40,41,44)(26,43,35,27)(29,45,46,34)(38,47,48,42);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,22,34,23,32,37,45)(18,25,27,42,28,41,43,47)(19,29,21,31,30,46,36,33)(24,38,35,40,39,48,26,44);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(20,32)(25,41)(29,46)(31,33)(34,45)(38,48)(40,44)(42,47);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,22,30,23,36,37,19)(18,26,27,24,28,35,43,39)(20,33,34,29,32,31,45,46)(25,40,42,48,41,44,47,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fh,d^2=g,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6747705);