Using multiple UAVs to manipulate the full posture of an object is a promising capability in many industrial applications, such as autonomous building construction and heavy-load transportation. Among various methods, manipulation via cables excels in mechanical simplicity and ease of use, but is challenging from a control perspective. Existing centralized control methods either neglect the dynamic coupling between UAVs and the load or resort to a cascade structure, which limits the operational speed and cannot guarantee safety. In this work, we propose a centralized control method that uses nonlinear model predictive control. This control method takes into account the full nonlinear model of the load-UAV system, as well as the constraints of UAV thrust, collision avoidance, and ensuring all cables are taut. By taking into account the above factors, the proposed control algorithm can fully exploit the performance of UAVs and facilitate the speed of operation. We demonstrate our algorithm through 6-DoF simulations to achieve fast and safe manipulation of the pose of a rigid-body payload using multiple UAVs.