Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#11602 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C2×D8 |
| commutator | C8 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C2×C5 | C2×D4⋊C8 |
| prime radical | C2×C5×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
380 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)(6,7,10,14,11,16,26,31,12,17,27,32,25,15,9,23)(8,18,20,13,22,29,36,30,19,34,35,28,21,37,24,33);(6,8)(7,13)(9,24)(10,20)(11,22)(12,19)(14,29)(15,33)(16,30)(17,28)(18,23)(21,25)(26,36)(27,35)(31,34)(32,37);(8,19)(13,28)(18,34)(20,35)(21,22)(24,36)(29,37)(30,33)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4590);