Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13039 |
| 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 | |
472 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)(10,11,16,22,26,41,14,33,18,24,15,35,17,23,34,21)(12,25,27,39,36,20,32,38,29,40,31,19,28,37,13,30);(6,7,8,9)(10,12,26,28,18,29,17,36)(11,19,23,30,24,39,41,38)(13,14,31,15,32,34,27,16)(20,22,40,21,37,35,25,33);(10,13)(11,20)(12,14)(15,28)(16,36)(17,27)(18,32)(19,22)(21,30)(23,40)(24,37)(25,41)(26,31)(29,34)(33,38)(35,39)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6027);