Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#23705 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
820 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)(10,11)(12,15)(13,14)(16,17)(18,19,23,28,24,29,22,27)(20,25,30,26,21,31,32,33);(6,7)(8,9)(11,14)(15,17)(18,20)(19,25)(21,24)(22,32)(23,30)(26,28)(27,33)(29,31);(7,9)(18,21,24,20)(19,26,29,33)(22,32,23,30)(25,27,31,28);(10,12,13,16)(11,15,14,17)(18,22,24,23)(19,27,29,28)(20,30,21,32)(25,26,31,33)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16693);