Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#29695 |
| 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 ◁ C22×C42×C5 ◁ 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)(11,12)(15,16)(18,19,22,26)(20,27,23,24)(21,29,30,33)(25,31,32,28);(6,7)(8,9)(10,11)(12,14)(13,15)(16,17)(18,20,22,23)(19,24,26,27)(21,28,30,31)(25,29,32,33);(7,9)(10,12,14,11)(13,15,17,16)(18,21,22,30)(19,25,26,32)(20,28,23,31)(24,29,27,33);(10,13,14,17)(11,15,12,16)(21,30)(25,32)(28,31)(29,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22683);