Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8626 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#8641 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #380 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #117 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,14,12,9,10,20)(8,15,18,16,19,21,17,13)(23,27)(24,31)(25,28)(30,35)(32,33)(34,36);(6,8,10,17,12,19,11,18)(7,13,14,15,9,16,20,21)(22,23,28,35,26,30,25,27)(24,29,31,36,33,37,32,34);(6,9)(7,12)(8,16)(10,20)(11,14)(13,19)(15,18)(17,21)(22,24,26,33)(23,29,30,37)(25,32,28,31)(27,34,35,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1629);