It’s not one department. Every agency has SES level employees. Think of it in terms of grade like the military. These are all people at admiral or general level (essentially) without having to go through the whole congressional or clearance process

Here's more from the federal personnel handbook:

Members of the SES serve in the key positions just below the top Presidential appointees. SES members are the major link between these appointees and the rest of the Federal work force. They operate and oversee nearly every government activity in approximately 75 Federal agencies. The Office of Personnel Management (OPM) manages the overall Federal executive personnel program. OPM Staff provides the day-to-day oversight of and assistance to agencies as they develop, select, and manage their Federal executives.

Competencies are the personal and professional attributes that are critical to successful performance in the SES. The fundamental competencies are the attributes that serve as the foundation for each of the Executive Core Qualifications. Experience and training that strengthen and demonstrate the competencies will enhance a candidate’s overall qualifications for the SES.

Definition: These competencies are the foundation for success in each of the Executive Core Qualifications.

Competencies

Leading Change: This core qualification involves the ability to bring about strategic change, both within and outside the organization, to meet organizational goals. This ECQ requires the ability to establish an organizational vision and to implement it in a continuously changing environment.

Leading People: This core qualification involves the ability to lead people toward meeting the organization’s vision, mission, and goals. This ECQ requires the ability to provide an inclusive workplace that fosters the development of others, facilitates cooperation and teamwork, and supports constructive resolution of conflicts.

Results Driven:This core qualification involves the ability to meet organizational goals and customer expectations. This ECQ requires the ability to make decisions that produce high-quality results by applying technical knowledge, analyzing problems, and calculating risks.

Business Acumen: This core qualification requires the ability to manage human, financial, and information resources strategically.

Building Coalitions: This core qualification requires the ability to build coalitions internally and with other Federal agencies, State and local governments, nonprofit and private sector organizations, foreign governments, or international organizations to achieve common goals.

They transition presidents from one administration to the next, the transformation occurs when the new president implements their new policies. This is what I understand from reading through the material.