Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11364 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #390 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,13)(9,11)(10,12)(14,15)(16,24,27,20)(17,28,31,21)(18,23)(19,22)(25,35)(26,40,32,36)(29,37)(30,41,33,38)(34,39)(42,43)(44,45);(6,8,10,9)(7,11,12,13)(14,16)(15,20)(17,29)(18,32)(19,27)(21,37)(22,24)(23,40)(25,30)(26,34)(28,45)(31,44)(33,43)(35,38)(36,39)(41,42);(6,9,10,8)(7,11,12,13)(14,17)(15,21)(16,25)(18,33)(19,31)(20,35)(22,28)(23,41)(24,42)(26,44)(27,43)(29,32)(30,34)(36,45)(37,40)(38,39)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4352);