Fix a finite group $G$. An $n$-cochain of $G$ with coefficients in a $G$-module $M$ is a function$$b:G^n\rightarrow M$$To determine $b$, one must specify the values $b(g_1,\ldots,g_n)\in M$ for all $(g_1,\ldots,g_n)\in G^n$, so one might say there are $|G|^n$ independent components of the $n$-cochain $b$.
Now let $b$ by an $n$-cocycle: that is, an $n$-cochain satisfying the constraint$$\delta b(g_1,\ldots,g_{n+1})=1$$where $\delta$ is the $n^\text{th}$ coboundary homomorphism.
The $|G|^n$ components of $b$ are no longer independent. Given an $n$-cocycle $b$, what is the minimum number of values of $b(g_1,\ldots,g_n)\in M$ that one must specify in order to determine it? Given only $G$, what is the supremum of this number over all $n$-cocycles of $G$?