#6533082

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

3234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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, 6440278);