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A finite group of order $n$ can be represented by its Cayley table. In theword-RAM model the Cayley table of a group of order $n$ can be stored using$O(n^2)$ words and can be used to answer a multiplication query in constanttime. It is interesting to ask if we can design a data structure to store agroup of order $n$ that uses $o(n^2)$ space but can still answer amultiplication query in constant time.  We design a constant query-time data structure that can store any finitegroup using $O(n)$ words where $n$ is the order of the group.  Farzan and Munro (ISSAC 2006) gave an information theoretic lower bound of$\Omega(n)$ on the number of words to store a group of order $n$. Since ourdata structure achieves this lower bound and answers queries in constant time,it is optimal in both space usage and query-time.  A crucial step in the process is essentially to design linear space andconstant query-time data structures for nonabelian simple groups. The datastructures for nonableian simple groups are designed using a lemma that weprove using the Classification Theorem for Finite Simple Groups (CFSG).

Linear Space Data Structures for Finite Groups with Constant Query-time