Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13250 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #442 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11)(12,22,25,38)(13,26,29,39)(14,30)(15,32)(16,21)(17,24,35,31)(18,28,36,40)(19,23)(20,33)(27,34)(37,41);(6,7,8,9)(10,12,23,40,33,29,14,31,16,25,41,28,15,13,27,24)(11,17,30,26,32,18,19,38,21,35,34,39,20,36,37,22);(10,13,27,40,15,25,41,24,16,29,14,28,33,12,23,31)(11,18,37,39,20,17,34,38,21,36,19,26,32,35,30,22)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6238);