  Let $\mathcal{C}_1$ denote the largest connected component of the critical
Erd\H{o}s--R\'{e}nyi random graph $G(n,{\frac{1}{n}})$. We show that,
typically, the diameter of $\mathcal{C}_1$ is of order $n^{1/3}$ and the mixing
time of the lazy simple random walk on $\mathcal{C}_1$ is of order $n$. The
latter answers a question of Benjamini, Kozma and Wormald. These results extend
to clusters of size $n^{2/3}$ of $p$-bond percolation on any $d$-regular
$n$-vertex graph where such clusters exist, provided that
$p(d-1)\le1+O(n^{-1/3})$.
