Regular orbits in powers of permutation representations

Let (Q, G) be a faithful permutation representation of a finite group G. Suppose that the G-set Q has t distinct non-zero marks. In a permutation representation analogue of a theorem of Brauer on linear representations. it is shown that the direct power (Q, G)(t) of (Q, G) contains a regular orbit. As a corollary, the probability that a random element of Q(r) lies in a regular orbit of (Q, G)(r) is shown to tend to I exponentially fast as r tends to infinity. Further, knowledge of the rate of convergence is equivalent to knowledge of the second largest value of the character of the linear permutation representation.