In this work we introduce a superconducting quantum processor architecture that uses a transmission-line resonator to implement effective all-to-all connectivity between six transmonqubits. This architecture can be used as a test-bed for algorithms that benefit from high connectivity. We show that the central resonator can be used as a computational element, which offers the flexibility to encode a qubit for quantum computation or to utilize its bosonic modes which further enables quantum simulation of bosonic systems. To operate the quantum processing unit (QPU), we develop and benchmark the qubit-resonator conditional Z gate and the qubit-resonator MOVE operation. The latter allows for transferring a quantum state between one of the peripheral qubits and the computational resonator. We benchmark the QPU performance and achieve a genuinely multi-qubit entangled Greenberger-Horne-Zeilinger (GHZ) state over all six qubits with a readout-error mitigated fidelity of 0.86.