By considering nondeterminism as a quantifiable resource, we introduce new nondeterministic complexity classes obtained from NC circuits using a bounded number of nondeterministic gates. Let NNC( f( n)) denote the class of languages computable by an NC circuit family with O( f( n)) nondeterministic gates. If f( n) is limited to log n, we show that the class obtained is equivalent to NC. If f( n) is allowed to encompass all polynomials, we show that the class obtained is equivalent to NP. The class of most interest, NNC(polylog), obtained by letting f( n) encompass all polylogarithmic functions, contains a version of the quasigroup (Latin square) isomorphism problem. The quasi-group isomorphism problem is not known to be in P or NP-complete; thus, NNC(polylog) is a candidate for separating NC and NP. We also show that NNC(polylog) ⊆ DSPACE(polylog). More specifically, we show that NNC k (log k n) is contained in DSPACE(log k n), where NNC k (log j n) denotes the complexity class obtained from an NC k circuit with O(log j n) nondeterministic gates. This containment yields DSPACE(log 2 n) algorithms for the quasigroup isomorphism problem, the Latin square isotopism problem and the Latin square graph isomorphism problem. The only previously known bound for these problems is Miller's time bound of n log2 n + O(1) . This result also generalizes the DSPACE(log 2 n) algorithm of Lipton (1976) for the group isomorphism problem. We also show that, for every k, NNC( n k ) ⊆ DSPACE( n k ), and if for some k there exists a j such that DSPACE( n k ) ⊆ NNC( n j ) then NP = PSPACE.