Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#13162 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C23×C5 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C16 ◁ G | |
290 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)(10,11,15,21,16,20,31,37,17,22,30,19,14,32,13,28)(12,18,25,33,26,34,23,38,27,35,39,36,24,40,29,41);(7,9)(10,12,15,25,16,26,31,23,17,27,30,39,14,24,13,29)(11,18,21,33,20,34,37,38,22,35,19,36,32,40,28,41);(10,13,14,30,17,31,16,15)(11,19,20,28,22,21,32,37)(12,23,24,25,27,29,26,39)(18,33,34,38,35,36,40,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6150);