Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #476 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
294 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)(12,22,26,40)(13,28)(14,18)(15,33)(16,21)(17,24,35,23)(19,30)(20,29)(25,38,27,39)(31,36,41,34)(32,37);(10,12,15,25,16,26,29,27)(11,17,20,34,21,35,33,36)(13,23,14,31,30,24,32,41)(18,22,28,39,37,40,19,38);(6,7,8,9)(10,13,29,32,16,30,15,14)(11,18,33,19,21,37,20,28)(12,23,27,41,26,24,25,31)(17,22,36,38,35,40,34,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6175);