Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#34182 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(7,9)(10,11,12,13)(15,16)(18,26)(20,22)(24,29);(14,15,19,16)(17,22,25,20)(18,21,26,23)(24,28,29,27);(6,7,8,9)(14,16,19,15)(17,22,25,20)(18,23,26,21)(24,28,29,27);(14,17,19,25)(15,20,16,22)(18,24,26,29)(21,27,23,28);(14,18)(15,21)(16,23)(17,24)(19,26)(20,27)(22,28)(25,29)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27170);