This question already has an answer here:
I am looking for an explicit description of the additive, multiplicative group structure and automorphism group of the field with 729 elements.
Sorry for not being clear. I have no clue what does "explicit description" mean, but I am guessing that the underlying group might be cyclic, but I do not know how to give a proof.
For its automorphism group, What I have tried: 1). Comparing this with the Galois group of this field over $F_3$ which is cyclic. 2). I also know that Aut(Z_3） = Z_2, which is, in general, true for or prime fields. That is why I am guessing that the underlying additive group should be cyclic, then by the same method, I can show that the automorphism group is a cyclic group of order 728 probably.