  The clique chromatic number of a graph is the minimum number of colors
required to assign to its vertex set so that no inclusion maximal clique is
monochromatic. McDiarmid, Mitsche and Pra\l at proved that the clique chromatic
number of the binomial random graph $G\left(n,\frac{1}{2}\right) $ is at most
$\left(\frac{1}{2}+o(1)\right)\log_2n$ with high probability. Alon and
Krivelevich showed that it is greater than $\frac{1}{2000}\log_2n$ with high
probability and suggested that the right constant in front of the logarithm is
$\frac{1}{2}.$ We prove their conjecture and, beyond that, obtain a tight
concentration result: whp $\chi_c\left(G\left(n,1/2\right)\right) =
\frac{1}{2}\log_2 n - \Theta\left(\ln\ln n\right).$
