Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#33443 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C22×D4 |
| commutator | C4 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×C8×D4 |
| prime radical | C2×C5×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×M32 ◁ G | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,16,10,17,25,34,12,14,27,30,26,15,31,33)(8,18,20,24,21,29,35,23,22,32,36,37,19,13,28,9);(6,8)(7,13)(9,16)(10,21)(11,20)(12,22)(14,29)(15,32)(17,18)(19,26)(23,30)(24,34)(25,35)(27,36)(28,31)(33,37);(6,9,11,18,10,24,25,29,12,23,27,32,26,37,31,13)(7,8,16,20,17,21,34,35,14,22,30,36,15,19,33,28);(7,14)(9,23)(13,29)(15,17)(16,30)(18,32)(24,37)(33,34)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26431);