Abstract: We study the links between the number n of vertices and the order a of the automorphism group a poset has. On the one hand a theorem gives the minimal value of n and the explicit structure of the minimal poset when a is fixed and prime. A corollary provides an upper bound when a is a prime power. On the other hand a table enumerates the nonisomorphic posets according to n=<12 and a. It specifies in which proportion the rigid posets are and shows that the corollary is sharp when a is a small power of 2.