Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#20459 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#117 ◁ G | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11,14,19,16,21,15,20)(12,22,24,31,25,35,29,17)(13,18,27,33,26,32,28,34)(23,36,38,40,37,41,39,30);(10,12,15,24,16,25,14,29)(11,17,20,22,21,31,19,35)(13,23,28,38,26,37,27,39)(18,30,34,36,32,40,33,41);(6,8,9,7)(10,13,16,26)(11,18,21,32)(12,23,25,37)(14,27,15,28)(17,30,31,40)(19,33,20,34)(22,36,35,41)(24,38,29,39);(13,26)(18,32)(23,37)(27,28)(30,40)(33,34)(36,41)(38,39)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13447);