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) = C2×C42 |
| direct product | C5×#13231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1964 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#131 ◁ G | |
468 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)(8,10,18,25)(9,19)(11,26)(12,15)(13,22)(14,20)(16,28,31,37)(17,32,29,24)(21,35,36,30)(23,33)(27,34);(6,8,10,15,12,18,25,7)(9,16,21,27,23,31,36,13)(11,17,24,26,20,29,32,14)(19,22,30,28,33,34,35,37);(6,9,20,34,12,23,11,22)(7,13,26,33,15,27,14,19)(8,16,29,35,18,31,17,30)(10,21,32,37,25,36,24,28)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6219);