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) = C2×C4 |
| direct product | C5×#13135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #459 |
| 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 | |
324 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)(6,7,9,8)(10,11)(12,20)(13,25)(14,30)(15,32)(16,21)(17,23)(18,27)(19,31)(22,36)(24,38)(26,40)(28,37)(29,41)(33,35)(34,39);(6,8,9,7)(10,12,14,23,16,24,31,15)(11,17,19,20,21,32,30,38)(13,22,26,39,29,27,28,33)(18,25,34,37,36,41,35,40);(10,13,14,26,16,29,31,28)(11,18,19,34,21,36,30,35)(12,22,23,39,24,27,15,33)(17,25,20,37,32,41,38,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6123);