#6564362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fg,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6471558);