Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,23)(22,24)(25,26,31,35)(27,32,36,40)(28,33,37,38)(29,30,34,39);(3,7)(5,8)(10,15)(12,22)(16,19)(18,21)(25,27,31,36)(26,32,35,40)(28,29,37,34)(30,38,39,33);(1,2)(3,5)(4,6)(7,8)(10,15)(11,20)(17,24)(18,21)(27,36)(28,37)(32,40)(33,38);(9,11)(10,16)(12,18)(13,17)(14,20)(15,19)(21,22)(23,24)(25,28)(26,33)(27,29)(30,32)(31,37)(34,36)(35,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,22)(10,17,15,24)(11,21,20,18)(13,16,23,19)(25,29)(26,30)(27,37)(28,36)(31,34)(32,38)(33,40)(35,39);(1,2)(3,5)(4,6)(7,8)(9,13,14,23)(10,18,15,21)(11,17,20,24)(12,19,22,16)(25,30)(26,34)(27,38)(28,32)(29,35)(31,39)(33,36)(37,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,e^2=g,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^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9040960);