Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#34602 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
2294 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)(8,9)(10,11)(12,15)(13,14)(16,17)(18,19)(20,23)(21,24)(22,25)(26,30)(27,31)(28,32)(29,33);(11,14)(15,17)(18,20,22,27)(19,23,25,31)(21,28,29,26)(24,32,33,30);(7,9)(18,21,22,29)(19,24,25,33)(20,26,27,28)(23,30,31,32);(19,25)(23,31)(24,33)(30,32);(10,12,13,16)(11,15,14,17)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27590);