#6555984

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1343 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,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, 6463180);