Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8790 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#8790 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #382 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,12)(8,13)(9,10)(14,15)(16,24,27,20)(17,28,30,43)(18,23)(19,22)(21,29,34,44)(25,38)(26,37,41,33)(31,35)(32,39)(36,45)(40,42);(6,7,10,8,11,13,9,12)(14,16,18,26,19,27,31,41)(15,20,23,33,22,24,35,37)(17,25,29,39,30,40,44,45)(21,32,28,42,34,36,43,38);(6,8,9,7,11,12,10,13)(14,17)(15,21,22,34)(16,25)(18,29)(19,30)(20,32,24,36)(23,28,35,43)(26,39)(27,40)(31,44)(33,42,37,38)(41,45)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1778);