Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#12563 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #106 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#106 |
| prime radical | C24×C5 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
570 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)(12,23,26,36)(13,19)(14,31)(15,32)(16,33)(17,22)(18,25,35,27)(20,30)(21,29)(24,37,28,41)(34,39,38,40);(7,9)(10,12,15,27,16,28,29,39,17,26,14,25,13,24,30,40)(11,18,21,36,19,34,32,41,22,35,20,23,33,38,31,37);(10,13,17,16)(11,19,22,33)(12,24,26,28)(14,29,15,30)(18,34,35,38)(20,31,21,32)(23,37,36,41)(25,39,27,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5551);