#6550370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533087
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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1527 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)(25,26,30,38,29,33,28,32)(27,34,35,31,37,39,36,40);(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22);(1,3,4,7)(2,6,5,8)(10,14)(15,21)(16,22)(23,24)(25,27)(26,31)(28,35)(29,37)(30,36)(32,39)(33,40)(34,38);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,28,29,30)(26,32,33,38)(27,35,37,36)(31,39,40,34);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)

Polycyclic presentation:

2^9|a^2=gi,c^2=f,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,c^b=ch,g^c=gi,e^d=eh

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6457566);