#6591335

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1285 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgi,c^2=fgi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6498531);