#6555967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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 ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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)(25,26,29,35,31,34,30,36)(27,37,40,32,28,39,38,33);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27,31,28)(26,32,34,33)(29,38,30,40)(35,37,36,39);(10,14)(15,21)(16,22)(23,24)(25,28,31,27)(26,33,34,32)(29,40,30,38)(35,39,36,37);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(26,34)(32,33)(35,36)(37,39);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29,31,30)(26,35,34,36)(27,38,28,40)(32,37,33,39)

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,d^a=di,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6463163);