Denote the set of these primes by $A$ and the number of elements in $A$ by n. There are 9 primes smaller than 27, namely $2,3,5,7,11,13,17,19$ and 23. Since 27 is odd and all primes except 2 are odd, $2 \notin A$. Thus the largest prime $p$ is at most $27 \cdot 9-3-5-7-11-13-17-19-23=145$, so $p \leq 141$. When the primes are $3,5,7,11,13,17,19,23,29,31$ and 139, their average is 27. Therefore $p=139$.