Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#22288 |
| 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 | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15)(16,19)(17,21)(18,20)(22,23,27,30,26,29,31,24)(25,33,35,32,34,28,36,37);(6,8)(7,10)(9,13)(11,12)(14,16,18,17)(15,19,20,21)(22,24,31,29,26,30,27,23)(25,32,36,33,34,37,35,28);(8,13)(10,12)(14,17,18,16)(15,19,20,21)(22,25,26,34)(23,28,29,33)(24,32,30,37)(27,35,31,36);(7,11)(10,12)(25,34)(28,33)(32,37)(35,36)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15276);