Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#23057 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
758 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)(8,9)(11,15)(16,17)(18,19,24,20,23,26,22,28)(21,25,29,27,30,33,31,32);(10,11)(12,16)(13,17)(14,15)(18,20,24,26,23,28,22,19)(21,27,29,33,30,32,31,25);(7,9)(10,12,14,13)(11,16,15,17)(18,21,23,30)(19,25,26,33)(20,27,28,32)(22,31,24,29);(10,13,14,12)(11,16,15,17)(18,22,23,24)(19,20,26,28)(21,29,30,31)(25,32,33,27)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16045);