Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,2)(3,8)(4,5)(6,7)(9,11,14,10)(12,16,13,15)(19,25)(22,28)(26,31)(29,32);(2,5)(6,8)(9,12,14,13)(10,15,11,16);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(18,23)(19,25)(24,30)(26,31)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=cfg,d^b=dfi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314898);