Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#17404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C42×C5 ◁ G | |
960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16,19,12)(9,13)(10,14)(11,15)(17,21,18,20)(22,23,26,29,27,30,25,34)(24,28,33,36,31,35,32,37);(6,8,11,19)(7,12,15,16)(9,17,10,18)(13,20,14,21)(22,24)(23,28)(25,32)(26,33)(27,31)(29,36)(30,35)(34,37);(6,9,11,10)(7,13,15,14)(8,17,19,18)(12,20,16,21)(24,31)(28,35)(32,33)(36,37);(6,10,11,9)(7,14,15,13)(8,18,19,17)(12,21,16,20)(22,25,27,26)(23,29,30,34)(24,32,31,33)(28,36,35,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10392);