Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#20441 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,8,9)(14,15,19,22,18,21,27,17)(16,23,26,24,25,20,28,29);(7,9)(10,11,12,13)(14,16)(15,20)(17,24)(18,25)(19,26)(21,23)(22,29)(27,28);(14,17,27,21,18,22,19,15)(16,24,28,23,25,29,26,20);(15,21)(16,25)(17,22)(26,28)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13429);