Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,25,21,23,20,29)(19,24,28,31,26,30,27,32);(1,2)(3,8)(4,5)(6,7)(11,16)(13,15)(18,23)(19,26)(25,29)(27,28);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,27)(21,26)(22,28)(23,30)(25,31)(29,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,21,22)(18,25,23,29)(19,27,26,28)(24,31,30,32);(19,26)(24,30)(27,28)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=dfi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314814);