  For a locally compact group G and a compact subgroup K, the corresponding
Hecke algebra consists of all continuous compactly supported complex functions
on G that are K-bi-invariant. There are many examples of totally disconnected
locally compact groups whose Hecke algebras with respect to the maximal compact
subgroups are not commutative. One of those is the universal group U(F)^+, when
F is primitive but not 2-transitive. For this class of groups we prove that the
Hecke algebra with respect to the maximal compact subgroup K is infinitely
generated and infinitely presented. This may be relevant for constructing
irreducible unitary representations of U(F)^+ whose subspace of K-fixed vectors
has dimension at least two, or answering the question whether U(F)^+ is a type
I group or not. On the contrary, when F is 2-transitive that Hecke algebra of
U(F)^+ is commutative, finitely generated admitting only one generator.
