The eigenvalue problem for the p-Laplace operator with p>1 on planar domains with the zero Dirichlet boundary condition is considered. The Constrained Descent Method and the Constrained Mountain Pass Algorithm are used in the Sobolev space setting to numerically investigate the dependence of the two smallest eigenvalues on p. Computations are conducted for values of p between 1.1 and 10. Symmetry properties of the second eigenfunction are also examined numerically. While for the disk an odd symmetry about the nodal line dividing the disk in halves is maintained for all the considered values of p, for rectangles and triangles symmetry changes as p varies. Based on the numerical evidence the change of symmetry in this case occurs at a certain value p_0 which depends on the domain.