Large language models (LLMs) are increasingly used for automated scoring of student essays. However, these models may perpetuate societal biases if not carefully monitored. This study analyzes potential biases in an LLM (XLNet) trained to score persuasive student essays, based on data from the PERSUADE corpus. XLNet achieved strong performance based on quadratic weighted kappa, standardized mean difference, and exact agreement with human scores. Using available metadata, we performed analyses of scoring differences across gender, race/ethnicity, English language learning status, socioeconomic status, and disability status. Automated scores exhibited small magnifications of marginal differences in human scoring, favoring female students over males and White students over Black students. To further probe potential biases, we found that separate XLNet classifiers and XLNet hidden states weakly predicted demographic membership. Overall, results reinforce the need for continued fairness analyses as use of LLMs expands in education.