'Randomness' of inverses of $(\mathbb{Z}/p \mathbb{Z})^\times$

Suppose you are given the group $(\mathbb{Z} / p \mathbb{Z})^{\times}$, where $p$ is prime. Let $A_p$ denote the sequence whose $j$th element is the inverse of $[j]$. For instance, if $p = 7$, the sequence $A_7$ is $(1,4,5,2,3,6)$. Suppose you truncate the sequence upto the $\alpha p$th term (where $\alpha$ is a very small constant compared to $1$). Then, as $p$ approaches $\infty$, is the set of all elements in this truncated sequence approximately uniform on $\{ 1,2,3,4, \ldots, p-1 \}$?

Note: This is NOT homework. I only want to gain some intuition.

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