Maximal connected grading classes of a finite-dimensional algebra
A
A
are in one-to-one correspondence with Galois covering classes of
A
A
which admit no proper Galois covering and therefore are key in computing the intrinsic fundamental group
π
1
(
A
)
\pi _1(A)
. Our first concern here are the algebras
A
=
M
n
(
C
)
A=M_n(\mathbb {C})
. Their maximal connected gradings turn out to be in one-to-one correspondence with the Aut
(
G
)
(G)
-orbits of non-degenerate classes in
H
2
(
G
,
C
∗
)
H^2(G,\mathbb {C}^*)
, where
G
G
runs over all groups of central-type whose orders divide
n
2
n^2
. We show that there exist groups of central-type
G
G
such that
H
2
(
G
,
C
∗
)
H^2(G,\mathbb {C}^*)
admits more than one such orbit of non-degenerate classes. We compute the family
Λ
\Lambda
of positive integers
n
n
such that there is a unique group of central-type of order
n
2
n^2
, namely
C
n
×
C
n
C_n\times C_n
. The family
Λ
\Lambda
is of square-free integers and contains all prime numbers. It is obtained by a full description of all groups of central-type whose orders are cube-free. We then establish the maximal connected gradings of all finite-dimensional semisimple complex algebras using the fact that such gradings are determined by dimensions of complex projective representations of finite groups. In some cases we give a description of the corresponding fundamental groups.