Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13232 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1964 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#131 ◁ G | |
448 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)(6,7,12,17,13,18,11,30)(8,19,23,26,24,34,22,37)(9,25,14,32,28,21,27,15)(10,16,20,33,29,36,31,35);(6,8,20,15)(7,14,33,26)(9,16,34,17)(10,21,12,23)(11,22,29,25)(13,24,31,32)(18,27,35,37)(19,30,28,36);(6,9,11,27,13,28,12,14)(7,15,17,25,18,32,30,21)(8,16,22,35,24,36,23,33)(10,26,20,34,29,37,31,19)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6220);