#6555982

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=i,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, 6463178);