A vertex triple $(u,v,w)$ of a graph is called a 2-geodesic if v is adjacent to both u and w and u is not adjacent to w. A graph is said to be 2-geodesic transitive if its automorphism group is transitive on the set of 2-geodesics. In this paper, a complete classification of 2-geodesic transitive graphs of order $p^n$ is given for each prime p and $n\le 3$. It turns out that all such graphs consist of three small graphs: the complete bipartite graph ${\mathsf{K}}_{4,4}$ of order 8, the Schläfli graph of order 27 and its complement, and fourteen infinite families: the cycles ${\mathbf{C}}_p,{\mathbf{C}}_{p^2}$ and ${\mathbf{C}}_{p^3}$, the complete graphs ${\mathsf{K}}_p,{\mathsf{K}}_{p^2}$ and ${\mathsf{K}}_{p^3}$, the complete multipartite graphs ${\mathsf{K}}_{p[p]}$, ${\mathsf{K}}_{p[p^2]}$ and ${\mathsf{K}}_{p^2[p]}$, the Hamming graph $H(2,p)$ and its complement, the Hamming graph $H(3,p)$, and two infinite families of normal Cayley graphs on the extraspecial group of order $p^3$ and exponent p.

顶点三元组$（你,v,w）$如果v与u和w 都相邻且u不与w相邻，则图的 称为 2测地线。如果一个图的自同构群在 2-测地线集合上是传递的，则称该图是 2-测地线传递的。本文给出了阶次2-测地线传递图的完整分类$p^n$对于每个素数p给出$n\le 3$。事实证明，所有这样的图都由三个小图组成：完全二分图${\mathsf{K}}_{4,4}$8 阶图、27 阶 Schläfli 图及其补图以及十四个无限族：循环${\mathbf{C}}_p,{\mathbf{C}}_{p^2}$和${\mathbf{C}}_{p^3}$，完整的图${\mathsf{K}}_p,{\mathsf{K}}_{p^2}$和${\mathsf{K}}_{p^3}$，完整的多部分图${\mathsf{K}}_{p[p]}$,${\mathsf{K}}_{p[p^2]}$和${\mathsf{K}}_{p^2[p]}$，汉明图$H（2,p）$及其补充，汉明图$H（3,p）$，以及超特殊阶群上的两个无限族正态凯莱图$p^3$和指数p。