#6574120

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2616 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481316);