Large Language Models (LLMs) have emerged as powerful tools capable of understanding and generating human-like text, offering transformative potential across diverse domains. The Security Operations Center (SOC), responsible for safeguarding digital infrastructure, represents one of these domains. SOCs serve as the frontline of defense in cybersecurity, tasked with continuous monitoring, detection, and response to incidents. However, SOCs face persistent challenges such as high alert volumes, limited resources, high demand for experts with advanced knowledge, delayed response times, and difficulties in leveraging threat intelligence effectively. In this context, LLMs can offer promising solutions by automating log analysis, streamlining triage, improving detection accuracy, and providing the required knowledge in less time. This survey systematically explores the integration of generative AI and more specifically LLMs into SOC workflow, providing a structured perspective on its capabilities, challenges, and future directions. We believe that this survey offers researchers and SOC managers a broad overview of the current state of LLM integration within academic study. To the best of our knowledge, this is the first comprehensive study to examine LLM applications in SOCs in details.