Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#20561 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C22×C5×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
990 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)(12,16)(13,15)(14,17)(18,19,22,21,23,25,30,29)(20,24,28,26,27,31,33,32);(6,7,8,9)(10,12)(11,14)(13,17)(15,16)(18,20)(19,24)(21,26)(22,28)(23,27)(25,31)(29,32)(30,33);(12,17)(14,16)(18,21,22,25,23,29,30,19)(20,26,28,31,27,32,33,24);(11,15)(14,16)(20,27)(24,31)(26,32)(28,33)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13549);