We introduce a novel representation named as the unified gripper coordinate space for grasp synthesis of multiple grippers. The space is a 2D surface of a sphere in 3D using longitude and latitude as its coordinates, and it is shared for all robotic grippers. We propose a new algorithm to map the palm surface of a gripper into the unified gripper coordinate space, and design a conditional variational autoencoder to predict the unified gripper coordinates given an input object. The predicted unified gripper coordinates establish correspondences between the gripper and the object, which can be used in an optimization problem to solve the grasp pose and the finger joints for grasp synthesis. We demonstrate that using the unified gripper coordinate space improves the success rate and diversity in the grasp synthesis of multiple grippers.