  Fault-tolerant quantum computation is a technique that is necessary to build
a scalable quantum computer from noisy physical building blocks. Key for the
implementation of fault-tolerant computations is the ability to perform a
universal set of quantum gates that act on the code space of an underlying
quantum code. To implement such a universal gate set fault-tolerantly is an
expensive task in terms of physical operations, and any possible shortcut to
save operations is potentially beneficial and might lead to a reduction in
overhead for fault-tolerant computations. We show how the automorphism group of
a quantum code can be used to implement some operators on the encoded quantum
states in a fault-tolerant way by merely permuting the physical qubits. We
derive conditions that a code has to satisfy in order to have a large group of
operations that can be implemented transversally when combining transversal
CNOT with automorphisms. We give several examples for quantum codes with large
groups, including codes with parameters [[8,3,3]], [[15,7,3]], [[22,8,4]], and
[[31,11,5]].
