Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#13105 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | C23 |
| commutator | C23 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle, prime radical | C23×C5 | C22×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ G | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,15)(13,14)(16,17)(18,19,23,27,24,20,28,30,22,26,31,33,21,29,25,32);(11,14)(15,17)(18,20,23,30,24,26,28,33,22,29,31,32,21,19,25,27);(7,9)(10,12,13,16)(11,15,14,17)(18,21,22,24)(19,20,26,29)(23,25,31,28)(27,30,33,32)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6093);