#6573836

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

2635 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)(11,16)(12,25)(13,18)(14,19)(15,20)(17,21)(22,37)(23,31)(24,32)(26,40)(27,33)(28,35)(29,36)(30,38)(34,39);(1,2,3,4)(9,11)(10,16)(12,22)(13,27)(14,21)(15,24)(17,19)(18,33)(20,32)(23,28)(25,37)(26,29)(30,39)(31,35)(34,38)(36,40);(6,8)(11,21)(12,24)(14,22)(16,17)(19,37)(23,27)(25,32)(26,39)(28,30)(31,33)(34,40)(35,38);(9,12,15,21)(10,17,20,25)(11,22,24,14)(13,23,29,39)(16,19,32,37)(18,34,36,31)(26,30,27,28)(33,38,40,35);(9,13,15,29)(10,18,20,36)(11,23,24,39)(12,26,21,27)(14,30,22,28)(16,31,32,34)(17,33,25,40)(19,38,37,35)

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 6481032);