#6603047

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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)(17,18,22,27)(19,28,30,24)(20,32,34,42)(21,26,23,37)(25,33,41,35)(29,44,45,48)(31,40,36,39)(38,46,43,47);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,40)(27,28)(32,48)(33,47)(34,45)(35,46)(37,39)(41,43)(42,44);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,20,22,34)(18,25,27,41)(19,29,30,45)(21,35,23,33)(24,38,28,43)(26,42,37,32)(31,47,36,46)(39,48,40,44);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,30)(24,28)(29,45)(31,36)(38,43)(39,40)(44,48)(46,47);(10,13)(11,15)(17,21,22,23)(18,26,27,37)(19,31,30,36)(20,33,34,35)(24,39,28,40)(25,32,41,42)(29,46,45,47)(38,44,43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=f,d^2=fi,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,e^c=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6510243);