  A graph with a trivial automorphism group is said to be rigid. Wright proved
that for $\frac{\log n}{n}+\omega(\frac 1n)\leq p\leq \frac 12$ a random graph
$G\in G(n,p)$ is rigid whp. It is not hard to see that this lower bound is
sharp and for $p<\frac{(1-\epsilon)\log n}{n}$ with positive probability
$\text{aut}(G)$ is nontrivial. We show that in the sparser case $\omega(\frac 1
n)\leq p\leq \frac{\log n}{n}+\omega(\frac 1n)$, it holds whp that $G$'s
$2$-core is rigid. We conclude that for all $p$, a graph in $G(n,p)$ is
reconstrutible whp. In addition this yields for $\omega(\frac 1n)\leq p\leq
\frac 12$ a canonical labeling algorithm that almost surely runs in polynomial
time with $o(1)$ error rate. This extends the range for which such an algorithm
is currently known.
