Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#20521 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle, prime radical | C23×C5 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(10,11)(12,16)(13,17)(14,15)(18,19,23,26,22,25,31,21)(20,27,30,29,28,24,32,33);(6,7,9,8)(11,15)(16,17)(18,20,22,28)(19,24,25,27)(21,29,26,33)(23,30,31,32);(6,8,9,7)(10,12,14,13)(11,16,15,17)(19,25)(20,28)(21,26)(30,32);(10,13,14,12)(11,17,15,16)(18,21,31,25,22,26,23,19)(20,29,32,27,28,33,30,24)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13509);