We analyze the necessary and sufficient conditions for exact recovery of the symmetric Latent Space Model (LSM) with two communities. In a LSM, each node is associated with a latent vector following some probability distribution. We show that exact recovery can be achieved using a semidefinite programming (SDP) approach. We also analyze when NP-hard maximum likelihood estimation is correct. Our analysis predicts the experimental correctness of SDP with high accuracy, showing the suitability of our focus on the Karush-Kuhn-Tucker (KKT) conditions and the second minimum eigenvalue of a properly defined matrix.