What is the name of the group of executive department heads who advise the President?
The name of the group of executive department heads who advise the President of the United States is the Cabinet.
The role of the Cabinet (established under Article 2 Section 2 of the Constitution- '...he 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...') is to advise the President on any matter pertaining to the duties of each member of the respective office.
The Cabinet includes the Vice President and the heads of 15 executive departments — the Secretaries of Agriculture, Commerce, Defense, Education, Energy, Health and Human Services, Homeland Security, Housing and Urban Development, Interior, Labor, State, Transportation, Treasury, and Veterans Affairs, as well as the Attorney General.
According to the US Constitution, a cabinet member cannot concurrently serve as an acting Governor or member of Congress. The President nominates his selected cabinet officer and presents the Senate with his choice: the Senate either supports or rejects this choice by a simple majority vote. If a majority vote supports the President's choice, the nominated cabinet member is sworn in and begins his/her duties after the swearing-in ceremony.
All members of the Cabinet are addressed as 'Secretary' with the exception of the United States Attorney General. The Secretary of State is the highest ranking Cabinet member; this Secretary is fourth in line to the Presidency, the others being the Vice President (1st in line), Speaker of the House (2nd in line), and President Pro Tempore of the Senate (3rd in line). Aside from the fifteen heads of executive departments listed above, five members hold 'Cabinet rank.' These are the US Trade Representative, the head of the Environmental Protection Agency, the head of the Office of National Drug Control Policy, the head of the Office of Management and Budget, and the White House Chief of Staff.
The Cabinet dates back to the first President of the United States, George Washington, who appointed a Cabinet of four people: Thomas Jefferson (the Secretary of State), Henry Knox (the Secretary of War), Edmund Randolph (the Attorney General), and Alexander Hamilton (Secretary of the Treasury).
Because Cabinet members are important successors in the presidential line, great care is taken that all members of the Cabinet will never be seen together in one location during ceremonial or national events.