Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#20757 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#117 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#382 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×#117 ◁ G | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16,18,12)(9,20,22,35)(10,15)(11,14)(13,21,26,36)(17,30)(19,29,33,25)(23,27)(24,32)(28,37)(31,34);(6,8,10,19,11,18,23,33)(7,12,15,25,14,16,27,29)(9,17,21,32,22,31,36,37)(13,24,20,34,26,28,35,30);(6,9)(7,13,14,26)(8,17)(10,21)(11,22)(12,24,16,28)(15,20,27,35)(18,31)(19,32)(23,36)(25,34,29,30)(33,37);(8,18)(12,16)(17,31)(19,33)(24,28)(25,29)(30,34)(32,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13745);