Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9000 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | #120 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
474 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)(7,9)(11,14,15,17)(18,19,22,26,24,27,23,32)(20,28,29,33,21,30,31,25);(6,7,8,9)(12,16)(14,17)(18,20,24,21)(19,25,27,33)(22,31,23,29)(26,30,32,28);(10,11)(12,17)(13,15)(14,16)(18,21,24,20)(19,25,27,33)(22,31,23,29)(26,28,32,30)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1988);