[SageMath][1] math says the genus of $G$ is $1$.

    sage: G = Graph([[0,1],[0,2],[1,2],[0,4],[0,5],[0,6],[0,7],[0,8],[0,9],[1,3],[1,5],[1,6],[1,7],[1,8],[1,9],[2,3],[2,4],[2,6],[2,7],[2,8],[2,9]])
    sage: G.genus()
    1
    sage: G.get_embedding()
     
    {0: [1, 2, 4, 5, 6, 7, 8, 9],
     1: [0, 9, 2, 3, 6, 5, 8, 7],
     2: [0, 7, 6, 3, 1, 9, 8, 4],
     3: [1, 2],
     4: [0, 2],
     5: [0, 1],
     6: [0, 1, 2],
     7: [0, 2, 1],
     8: [0, 1, 2],
     9: [0, 2, 1]} 

   

  [1]: http://doc.sagemath.org/html/en/reference/graphs/sage/graphs/generic_graph.html