#6624658

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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=i,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6531854);