Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#22276 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#493 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ G | |
1050 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)(6,7,9,8)(10,11)(12,15)(13,16)(14,17)(18,19)(20,22)(21,28)(23,27)(24,30)(25,31)(26,32)(29,33);(6,8,9,7)(10,12,15,11)(13,16,14,17)(18,20,22,19)(21,24,29,26)(23,27,31,25)(28,32,33,30);(10,13,15,14)(11,16,12,17)(18,21,23,30,22,29,31,32)(19,24,25,28,20,26,27,33);(10,14,15,13)(11,16,12,17)(19,20)(24,26)(25,27)(28,33)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15264);