Given a finite group
G
\mathrm {G}
, the faithful dimension of
G
\mathrm {G}
over
C
\mathbb {C}
, denoted by
m
f
a
i
t
h
f
u
l
(
G
)
m_\mathrm {faithful}(\mathrm {G})
, is the smallest integer
n
n
such that
G
\mathrm {G}
can be embedded in
G
L
n
(
C
)
\mathrm {GL}_n(\mathbb {C})
. Continuing the work initiated by Bardestani et al. [Compos. Math. 155 (2019), pp. 1618–1654], we address the problem of determining the faithful dimension of a finite
p
p
-group of the form
G
R
≔
exp
(
g
R
)
\mathscr {G}_R≔\exp (\mathfrak {g}_R)
associated to
g
R
≔
g
⊗
Z
R
\mathfrak {g}_R≔\mathfrak {g}\otimes _\mathbb {Z}R
in the Lazard correspondence, where
g
\mathfrak {g}
is a nilpotent
Z
\mathbb {Z}
-Lie algebra and
R
R
ranges over finite truncated valuation rings.
Our first main result is that if
R
R
is a finite field with
p
f
p^f
elements and
p
p
is sufficiently large, then
m
f
a
i
t
h
f
u
l
(
G
R
)
=
f
g
(
p
f
)
m_\mathrm {faithful}(\mathscr {G}_R)=fg(p^f)
where
g
(
T
)
g(T)
belongs to a finite list of polynomials
g
1
,
…
,
g
k
g_1,\ldots ,g_k
, with non-negative integer coefficients. The latter list of polynomials is uniquely determined by the Lie algebra
g
\mathfrak {g}
. Furthermore, for each
1
≤
i
≤
k
1\le i\leq k
the set of pairs
(
p
,
f
)
(p,f)
for which
g
=
g
i
g=g_i
is a finite union of Cartesian products
P
×
F
\mathscr P\times \mathscr F
, where
P
\mathscr P
is a Frobenius set of prime numbers and
F
\mathscr F
is a subset of
N
\mathbb N
that belongs to the Boolean algebra generated by arithmetic progressions. Previously, existence of such a polynomial-type formula for
m
f
a
i
t
h
f
u
l
(
G
R
)
m_\mathrm {faithful}(\mathscr {G}_R)
was only established under the assumption that either
f
=
1
f=1
or
p
p
is fixed.
Next we formulate a conjectural polynomiality property for the value of
m
f
a
i
t
h
f
u
l
(
G
R
)
m_\mathrm {faithful}(\mathscr {G}_R)
in the more general setting where
R
R
is a finite truncated valuation ring, and prove special cases of this conjecture. In particular, we show that for a vast class of Lie algebras
g
\mathfrak {g}
that are defined by partial orders,
m
f
a
i
t
h
f
u
l
(
G
R
)
m_\mathrm {faithful}(\mathscr {G}_R)
is given by a single polynomial-type formula.
Finally, we compute
m
f
a
i
t
h
f
u
l
(
G
R
)
m_\mathrm {faithful}(\mathscr {G}_R)
precisely in the case where
g
\mathfrak {g}
is the free metabelian nilpotent Lie algebra of class
c
c
on
n
n
generators and
R
R
is a finite truncated valuation ring.