Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#12567 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C24×C5 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
586 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)(13,19)(14,30)(15,20)(16,33)(17,22)(18,25)(21,32)(24,35)(26,34)(27,41)(28,37)(29,38)(31,36)(39,40);(10,12,15,26,16,27,32,25,17,28,29,40,13,24,14,31)(11,18,21,35,19,34,20,37,22,36,30,41,33,39,38,23);(10,13,17,16)(11,19,22,33)(12,24,28,27)(14,29,32,15)(18,34,36,39)(20,30,38,21)(23,35,37,41)(25,26,31,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5555);