#6651627

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632048
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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26,29,34)(27,36,38,31)(28,35,30,33)(32,39,40,37);(2,5)(3,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27)(26,31)(28,39)(29,38)(30,37)(32,35)(33,40)(34,36);(1,3,4,7)(2,6,5,8)(25,27)(26,32)(28,37)(29,38)(30,39)(31,35)(33,36)(34,40);(2,5)(6,8)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,28,29,30)(26,33,34,35)(27,37,38,39)(31,32,36,40);(10,16)(14,17)(15,20)(23,24)(25,28,29,30)(26,33,34,35)(27,37,38,39)(31,32,36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=h,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6558823);