#8973299

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8880495);