Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9345 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
414 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)(6,7,9,12,11,15,10,14)(8,16,17,13,18,20,19,21)(22,23,26,31,27,30,25,29)(24,28,32,35,33,36,34,37);(7,12,15,14)(13,20,21,16)(22,24,25,32,27,33,26,34)(23,28,31,37,30,36,29,35);(6,8,11,18)(7,13,15,21)(9,17,10,19)(12,20,14,16)(22,25,27,26)(23,29,30,31)(24,32,33,34)(28,35,36,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2340);