Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#29777 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(11,14)(15,17)(18,19,23,27,22,26,31,33)(20,28,29,25,21,24,30,32);(6,7)(8,9)(10,11,13,14)(12,15,16,17)(18,20,22,21)(19,24,26,28)(23,29,31,30)(25,27,32,33);(7,9)(10,12)(11,15)(13,16)(14,17)(18,21,22,20)(19,25,26,32)(23,29,31,30)(24,27,28,33);(12,16)(15,17)(20,21)(24,28)(25,32)(29,30)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22765);