Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9781 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#9782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #390 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,10,12,13)(9,11)(14,15)(16,21)(17,26)(18,28)(19,23)(20,24)(22,27)(25,29);(6,8)(7,10,11,13)(9,12)(14,16,18,24,19,25,27,17)(15,20,22,21,23,26,28,29);(7,11)(10,13)(14,17,27,25,19,24,18,16)(15,21,28,20,23,29,22,26)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2770);