The Nestene High Command was presumably the leader of the Nestene Consciousness, who had been invading planets for a thousand million years. (TV: Spearhead from Space)

During the Second Nestene invasion of Earth, the Auton Leader spoke for the Nestene High Command in their alliance with the Master, and warned him that the High Command would be displeased with him and punish him should the invasion fail. (PROSE: Doctor Who and the Terror of the Autons)

As the Nestenes are a hive mind and singular entity, it is unclear who or what the High Command is, its only mention being in the novelisation of Terror of the Autons.
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