Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8496 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
676 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)(7,11,13,16)(12,20,21,19)(22,23)(24,32,35,37)(25,28)(26,30)(27,31)(29,34,36,33);(6,7)(8,12)(9,16)(10,13)(11,18)(14,19)(15,21)(17,20)(23,28)(25,26)(29,32)(30,31)(33,34)(36,37);(6,8,10,15)(7,12,13,21)(9,17,18,14)(11,19,16,20)(22,24)(23,29)(25,34)(26,33)(27,35)(28,32)(30,37)(31,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1484);