Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8803 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#8804 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #383 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #126 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,15,17,19)(9,11)(10,14)(12,16,20,21)(13,18)(22,23)(24,32,34,37)(25,29,35,28)(26,30)(27,31)(33,36);(7,11,14,13)(12,15,20,19)(22,24,33,35,27,34,26,25)(23,28,30,32,31,29,36,37);(6,8,9,16,10,17,18,21)(7,12,13,19,14,20,11,15)(22,25,26,34,27,35,33,24)(23,29,30,37,31,28,36,32)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1792);