#6591193

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=fgi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6498389);