Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#20564 |
| 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 | |
970 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)(18,19,23,21,22,25,32,31)(20,26,29,27,28,24,33,30);(6,7,8,9)(10,12)(11,14)(13,17)(15,16)(18,20)(19,24)(21,30)(22,28)(23,29)(25,26)(27,31)(32,33);(12,17)(14,16)(18,21,23,25,22,31,32,19)(20,27,29,24,28,30,33,26);(11,15)(14,16)(20,28)(24,26)(27,30)(29,33)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13552);