The Constitution actually says very little about the President's cabinet (the word "cabinet" is not even used). All it does is sort of imply that the President has power.
The most important wording in the Constitution on this subject is
[The President] may require the Opinion, in writing, of the principal Officer in each of the executive Departments, upon any subject relating to the Duties of their respective Offices.
In addition, the Constitution says that the President shall take care that the laws be faithfully executed and that the President can appoint government ministers. Between them, these provisions imply that the President can have a cabinet and that he (no women yet) is in charge of that cabinet.
Constitutionally, then, the President has the power to appoint the cabinet members (with the advice and consent of the Senate). This means that he can also fire them. He can require them to give him their opinion on things. Beyond that, the Constitution says nothing.