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) = C28 |
| direct product | C5×#34328 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3,4,5)(6,7,9,8)(10,11,14,13)(12,16,17,15)(19,20)(22,30)(24,26)(28,33);(18,19,23,20)(21,26,29,24)(22,25,30,27)(28,32,33,31);(10,12,14,17)(11,15,13,16)(18,20,23,19)(21,26,29,24)(22,27,30,25)(28,32,33,31);(10,11,14,13)(12,15,17,16)(18,21)(19,24)(20,26)(22,28)(23,29)(25,31)(27,32)(30,33);(6,8,9,7)(10,13,14,11)(12,16,17,15)(18,22)(19,25)(20,27)(21,28)(23,30)(24,31)(26,32)(29,33)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27316);