Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#22557 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
990 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)(7,9)(10,11)(12,14)(13,15)(16,17)(18,19,22,25,21,23,28,30)(20,24,27,31,26,29,32,33);(6,7)(8,9)(10,12,13,16)(11,14,15,17)(19,23)(24,29)(25,30)(31,33);(11,15)(14,17)(18,20)(19,24)(21,26)(22,27)(23,29)(25,31)(28,32)(30,33);(7,9)(20,26)(24,29)(27,32)(31,33)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15545);