#6683264

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6590460);