The modern notion of the essential dimension of a group gives a precise way to state your question (and generalizations), and there is some recent work extending the work mentioned in Scott's answer. To get started, see the article

J. Buhler and Z. Reichstein, On the essential dimension of a group, Compositio Math. 106 (1997), 159-179.

For instance, it is proved there that for polynomials of degree $n$, at least $\lfloor n/2 \rfloor$ coefficients are required. (This agrees with what you mentioned for $n=5$ and $n=6$.)

The modern notion of the essential dimension of a group gives a precise way to state your question (and generalizations), and there is some recent work extending the work mentioned in Scott's answer. To get started, see the article

J. Buhler and Z. Reichstein, On the essential dimension of a group, Compositio Math. 106 (1997), 159-179.

For instance, it is proved there that for polynomials of degree $n$, at least $\lfloor n/2 \rfloor$ coefficients are required. (This agrees with what you mentioned for $n=5$ and $n=6$.)