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The site-symmetry group of a point $\mathbf{r}$ in a crystal structure is the subgroup of its space group $G$ that fixes that point:It is the stabilizer subgroup for the action on points. Although the space group contains infinitely many lattice translations, the site-symmetry group is finite and is isomorphic to a subgroup of the corresponding crystallographic point group.^[ref]

For an original example, take a lattice with inversion symmetry and no other point operations, the space-group type conventionally denoted $P\bar{1}$. At the origin, both the identity and inversion $\mathbf{r}\mapsto -\mathbf{r}$ fix the site, giving a group of order2. A generic site has only the identity in its site-symmetry group; inversion instead produces a distinct site at $-\mathbf{r}$.

The equality in the definition fixes the actual point. At $\mathbf{r}=\mathbf{a}/2$, inversion alone moves it to $-\mathbf{a}/2$. Inversion followed by translation by $\mathbf{a}$ fixes it. Reducing fractional coordinates modulo integers can conceal that translation, but the full space group operation retains it.

The crystallographic site multiplicity consequently satisfiesIn this primitive inversion example the general multiplicity is2, so the origin has multiplicity1 while a generic site has multiplicity2. This is why high site symmetry reduces the number of distinct sites generated in a cell.^[ref]

A Wyckoff position of a space group consists of all points whose site-symmetry groups are conjugate subgroups of that space group. It classifies sites by their symmetry relationship within the whole group. A Wyckoff letter labels each such class in the conventional tables; the label is accompanied by its crystallographic site multiplicity.^[ref]

A Wyckoff position need not specify one particular atom or one fixed coordinate triplet. A general position usually has free fractional parameters. Choosing those parameters produces a crystallographic orbit, the collection of equivalent sites generated from that choice by the space group. Different parameter choices can give distinct orbits belonging to the same Wyckoff position.

An original comparison uses $P\bar{1}$, with operations $\mathbf{r}\mapsto \pm \mathbf{r}+\mathbf{n}$, where $\mathbf{n}$ is a lattice vector. Generic positions form pairs $\{\mathbf{r},-\mathbf{r}\}$ modulo lattice translations and have trivial site-symmetry groups. Many different pairs therefore belong to the same general Wyckoff position.

The origin and the half-cell point $(1/2,0,0)$ instead each have an inversion-containing site-symmetry group of order2. Their groups are isomorphic but not conjugate within this space group: a group operation sends the origin to an integer lattice point, never to that half-cell point. They occupy different special Wyckoff positions. Group order or multiplicity alone therefore does not uniquely identify a Wyckoff position.

This classification helps constrain independent atomic coordinates in a crystal structure; it describes allowed symmetry, not which chemical species occupies a site or its fractional occupation.