Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13282 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
414 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)(12,16)(13,14)(15,17)(19,25)(20,21)(22,23)(26,31)(27,28)(29,32)(30,33);(7,9)(11,14)(15,16)(18,19,26,32,23,29,21,25,24,30,20,27,22,28,31,33);(10,12,13,17)(11,15,14,16)(18,20,23,31,24,26,22,21)(19,27,29,33,30,32,28,25)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6270);