If I pick a random regular graph on $n$-vertices and degree $d$ from uniform distribution what is the probability that its automorphism group is of size at least $m$?

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I want to know what is the largest $\alpha>0$ such that there is a $d$ regular graph $G$ on $n$ vertices with automorphism group larger than ${(n-\alpha)!}$. Are such graphs efficiently constructible and how many such non-isomorphic graphs are possible?