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) = C26 |
| direct product | C5×#20673 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#117 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#380 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×M16 ◁ G | |
1292 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)(6,7,11,15,12,16,10,25)(8,17,21,28,19,32,20,13)(9,22,23,29,24,14,26,30)(18,33,35,37,34,31,36,27);(6,8,10,20,12,19,11,21)(7,13,15,17,16,28,25,32)(9,18,23,35,24,34,26,36)(14,27,29,31,22,37,30,33);(6,9)(7,14)(8,18)(10,23)(11,26)(12,24)(13,27)(15,29)(16,22)(17,31)(19,34)(20,35)(21,36)(25,30)(28,37)(32,33);(8,19)(13,28)(17,32)(18,34)(20,21)(27,37)(31,33)(35,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13661);