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) = C23×C4 |
| direct product | C5×#22055 |
| 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×#449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1024 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)(12,16)(13,15)(14,17)(18,19)(20,30)(21,25)(22,29)(23,28)(24,27)(26,32)(31,33);(6,8,9,7)(10,12)(11,14)(13,17)(15,16)(18,20)(19,25)(21,23)(22,32)(24,31)(26,27)(28,30)(29,33);(12,17)(14,16)(18,21,23,20)(19,26,28,33)(22,32,24,31)(25,29,30,27);(11,15)(14,16)(18,22,23,24)(19,27,28,29)(20,31,21,32)(25,33,30,26)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15043);