Group theory provides the conceptual framework for solving mathematical measurement problems in various fields such as computer science, mathematics, science and statistics, thus widely used in many researches. However, while performing with binary operation between group elements, solving the Binary Optimization-based Set Union Knapsack Problem (BO-SUKP) is more challenging in group theory since, the axioms (binary, associative, identity, inverse, and commutativity) of groups are not satisfied, as the problem involves selecting a subset of items from multiple sets to maximize a certain objective function while satisfying certain constraints. Hence, a novel Gradient Group Optimizer based Corona Product with Meta-Heuristic for Set-Union Knapsack Problem in Group Theory is introduced to tackle the aforementioned issues. In this novel approach, the Gradient group optimizer includes union sets to resolve the Binary Optimization (BO) variations in group theory and then Corona Product with Metaheuristic approach is developed to solve the Set Union Knapsack Problem (SUKP) by splitting up the union set's components. This proposed approach effectively solves the binary optimization issues in the set union knapsack problem over group theory by precisely satisfying the group axioms. Furthermore, the proposed model showed that the solutions are effective with best class, worst class, mean, and standard deviation.