#6533272

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3306 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440468);