Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#34085 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
2094 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)(10,11)(12,15)(13,14)(16,17)(18,19)(20,27)(21,25)(22,31)(23,26)(24,29)(28,33)(30,32);(6,8,9,7)(18,20)(19,24)(21,30)(22,28)(23,29)(25,33)(26,27)(31,32);(11,14)(15,17)(18,21,23,22)(19,25,26,31)(20,28,29,30)(24,32,27,33);(18,22,23,21)(19,25,26,31)(20,28,29,30)(24,33,27,32);(10,12,13,16)(11,15,14,17)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27073);