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×#8500 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3546 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
736 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)(8,15,17,11)(9,14)(10,13)(12,16,20,18)(19,21)(22,23,26,29,27,31,25,30)(24,32,33,28,34,36,35,37);(6,8)(7,11)(9,16)(10,17)(12,14)(13,15)(18,19)(20,21)(22,24)(23,28)(25,35)(26,33)(27,34)(29,36)(30,32)(31,37);(6,8)(7,12)(9,18)(10,17)(11,14)(13,20)(15,21)(16,19)(23,29)(25,26)(28,36)(30,31)(32,37)(33,35)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1488);