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×#34286 |
| 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 | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3,4,5)(6,7,9,8)(11,14)(15,17)(18,19)(20,25)(21,26)(22,30)(23,24)(27,31)(28,29)(32,33);(6,7,9,8)(19,24)(21,25)(27,30)(29,32);(6,8,9,7)(10,11)(12,15)(13,14)(16,17)(18,20,23,26)(19,25,24,21)(22,28,31,33)(27,32,30,29);(18,21,23,25)(19,26,24,20)(22,29,31,32)(27,33,30,28);(10,12,13,16)(11,15,14,17)(18,22)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(26,33)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27274);