Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#12596 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | #432 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
310 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)(10,11,15,20,14,22,30,37,16,21,32,18,31,19,13,28)(12,23,26,38,25,17,24,33,27,35,39,41,40,34,29,36);(6,7,8,9)(10,12)(11,17,21,34)(13,29)(14,25)(15,26)(16,27)(18,36,20,33)(19,23,22,35)(24,30)(28,38,37,41)(31,40)(32,39);(10,13,14,15,16,30,31,32)(11,18,19,37,21,20,22,28)(12,24,25,39,27,29,40,26)(17,33,23,38,34,36,35,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5584);