Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#21601 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | C23 |
| 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×C5×C8 ◁ G | |
862 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)(10,11)(12,17)(13,15)(14,16)(18,19,23,26,22,25,31,21)(20,27,30,29,28,24,32,33);(10,12,14,13)(11,15,16,17)(18,20,22,28)(19,24,25,27)(21,29,26,33)(23,30,31,32);(7,9)(20,28)(24,27)(29,33)(30,32);(10,13,14,12)(11,15,16,17)(18,21,31,25,22,26,23,19)(20,29,32,27,28,33,30,24)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14589);