This paper investigates the coordination of nonlinear sampled-data multi-agent systems subject to data rate constraint. The purpose is to design resource-efficient communication and control strategies that guarantee exponential synchronization. Two implementation scenarios are considered, the period time-triggered control and the period event-triggered control. One of the main difficulties of the problem is to obtain an explicit formula for the maximum allowable sampling period (MASP). To this end, an approach on finding the MASP for periodic time-triggered control is proposed first. Then, an asynchronous period event-triggered control strategy is formulated, a communication function and a control function are designed for each agent to determine respectively whether or not the sampled data and the control input should be transmitted at each sampling instant. Finally, the constraint of limited data rate is considered. An observer-based encoder-decoder and a finite-level quantizer are designed respectively for the Sensor-Controller communication and the Controller-Actuator communication such that certain constraint on the data rate is satisfied. It is shown that exponential synchronization can still be achieved in the presence of data rate constraint. A simulation example is given to illustrate the effectiveness of the theoretical results