  Typical measures of nonstabilizerness of a system of $N$ qubits require
computing $4^N$ expectation values, one for each Pauli string in the Pauli
group, over a state of dimension $2^N$. For permutationally invariant systems,
this exponential overhead can be reduced to just $O(N^3)$ expectation values on
a state with a dimension $O(N)$. We exploit this simplification to study the
nonstabilizerness phase transitions of systems with hundreds of qubits.
