Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11280 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #393 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1442 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,15,11,14,25,31)(8,16,19,32,18,12,24,27)(9,20,23,34,22,13,28,30)(17,26,33,36,21,29,35,37);(6,8)(7,12)(9,21,22,17)(10,24)(11,18)(13,29,20,26)(14,16)(15,32)(19,25)(23,33,28,35)(27,31)(30,36,34,37);(6,9)(7,13,14,20)(8,17)(10,23)(11,22)(12,26,16,29)(15,30,31,34)(18,21)(19,33)(24,35)(25,28)(27,36,32,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4268);