#6603033

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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)(17,18,23,27)(19,24,29,39)(20,32,34,43)(21,26,22,36)(25,37,42,35)(28,44,33,48)(30,40,31,46)(38,47,41,45);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,29)(25,41)(26,40)(27,39)(28,34)(32,48)(35,47)(36,46)(37,45)(38,42)(43,44);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,20,23,34)(18,25,27,42)(19,28,29,33)(21,35,22,37)(24,38,39,41)(26,43,36,32)(30,45,31,47)(40,48,46,44);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,29)(20,34)(24,39)(25,42)(30,31)(32,43)(35,37)(40,46);(10,13)(11,15)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,35,34,37)(24,40,39,46)(25,43,42,32)(28,45,33,47)(38,48,41,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6510229);