We present the conceptual design of GMagAO-X, an extreme adaptive optics system for the 25 m Giant Magellan Telescope (GMT). We are developing GMagAO-X to be available at or shortly after first-light of the GMT, to enable early high contrast exoplanet science in response to the Astro2020 recommendations. A key science goal is the characterization of nearby potentially habitable terrestrial worlds. GMagAO-Xis a woofer-tweeter system, with integrated segment phasing control. The tweeter is a 21,000 actuator segmented deformable mirror, composed of seven 3000 actuator segments. A multi-stage wavefront sensing system provides for bootstrapping, phasing, and high order sensing. The entire instrument is mounted in a rotator to provide gravity invariance. After the main AO system, visible (g to y) and near-IR (Y to H) science channels contain integrated coronagraphic wavefront control systems. The fully corrected and, optionally, coronagraphically filtered beams will then be fed to a suite of focal plane instrumentation including imagers and spectrographs. This will include existing facility instruments at GMT via fiber feeds. To assess the design we have developed an end-to-end frequencydomain modeling framework for assessing the performance of GMagAO-X. The dynamics of the many closed-loop feedback control systems are then modeled. Finally, we employ a frequency-domain model of post-processing algorithms to analyze the final post-processed sensitivity. The CoDR for GMagAO-X was held in September, 2021. Here we present an overview of the science cases, instrument design, expected performance, and concept of operations for GMagAO-X.