Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13466 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #2065 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
314 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)(6,7,14,19)(8,20)(9,26)(10,17,31,37)(11,16,12,34)(13,21,33,18)(15,28)(22,30)(23,36)(24,27)(25,35)(29,32);(6,8,10,22,13,24,11,32,14,25,31,26,33,15,12,23)(7,9,16,20,18,29,17,28,19,30,34,35,21,36,37,27);(6,9)(7,15)(8,21)(10,28)(11,20)(12,35)(13,36)(14,30)(16,22)(17,23)(18,25)(19,24)(26,34)(27,31)(29,33)(32,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6454);