Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#14351 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
1422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(11,14)(15,17)(18,19,22,20)(21,23,25,24)(26,27,30,29)(28,32,33,31);(19,20)(23,24)(28,33)(31,32);(6,7,8,9)(10,11,13,14)(12,15,16,17)(18,20,22,19)(21,23,25,24)(26,28,30,33)(27,31,29,32);(10,12,13,16)(11,15,14,17)(18,21)(19,23)(20,24)(22,25)(26,29,30,27)(28,32,33,31)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 7339);