My Group Theory notes give as definition for an ascending series:
$G_0, G_1,...,G_n \unlhd G $ such that $\{1\} = G_0 \unlhd G_1 \unlhd ... \unlhd G_n=G$ and $G_{i+1}/ G_{i} \subset Z(G/G_{i})$. As an example the series for $D_8$ (dihedral group of order 8) is given:
$$\{1\} \unlhd Z(D_8) \unlhd D_8$$
I don't see how this satisfies the definition at all. Computing $Z(D_8/Z(D_8))$ gives you
$$Z(D_8/Z(D_8))=Z(D_8)/Z(D_8)= {1}\neq D_8/Z(D_8),$$
where we use that $Z(D_8) \subset Z(D_8)$ and the correspondence theorem. What am I missing?