Let (X,x) be a pointed space. There is an action of π_{1}(X,x) on π_{n}(X,x) -- determined by considering π_{n}(X,x)=π_{n-1}(Ω_{x}X,**x**), where Ω_{x}X denotes the space of loops in X based at x, and **x** denotes the constant loop -- given simply by conjugation. We can speak unambiguously of π_{n}(X), the *free* (i.e., not necessarily basepoint-preserving) homotopy group exactly when this action is trivial.

On an algebraic level I'm fine with this, but I'm having trouble envisioning how a homotopy class might be conjugated to a different homotopy class in this way. Besides my admittedly small collection of toy examples, my issue could also be that I'm mainly thinking about π_{1}, in which case it might (???) be that the action *is* trivial. (I seem to recall that before learning about general homotopy theory, I heard a statement along the lines of "for path-connected spaces, you may as well ignore basepoints". Certainly the groups are all isomorphic, but I'm not certain whether there is a unique natural isomorphism.)

Also, are there (necessary and/or sufficient) conditions for when the π_{1} action on π_{n} will or won't be trivial, and does this depend on n?