Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12820 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #459 |
| 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×C4×C5×C8 ◁ G | |
414 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)(6,7,8,9)(10,11)(12,20)(13,25)(14,29)(15,31)(16,21)(17,23)(18,22)(19,30)(24,38)(26,37)(27,41)(28,33)(32,34)(35,39)(36,40);(10,12,14,23,16,24,30,15)(11,17,19,20,21,31,29,38)(13,22,26,39,28,40,27,32)(18,33,34,37,36,25,35,41);(10,13)(11,18)(12,22)(14,26)(15,32)(16,28)(17,33)(19,34)(20,37)(21,36)(23,39)(24,40)(25,31)(27,30)(29,35)(38,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5808);