  The group theoretical description of the periodic system of elements in the
framework of the Rumer-Fet model is considered. We introduce the concept of a
single quantum system, the generating core of which is an abstract
$C^\ast$-algebra. It is shown that various concrete implementations of the
operator algebra depend on the structure of generators of the fundamental
symmetry group attached to the energy operator. In the case of generators of
the complex shell of a group algebra of a conformal group, the spectrum of
states of a single quantum system is given in the framework of the basic
representation of the Rumer-Fet group, which leads to a group-theoretic
interpretation of the Mendeleev's periodic system of elements. A mass formula
is introduced that allows to give the termwise mass splitting for the main
multiplet of the Rumer-Fet group. The masses of elements of the Seaborg table
(eight-periodic extension of the Mendeleev table) are calculated starting from
the atomic number $Z=121$ to $Z=220$. The continuation of Seaborg homology
between lanthanides and actinides is established to the group of
superactinides. A 10-periodic extension of the periodic table is introduced in
the framework of the group-theoretic approach. The multiplet structure of the
extended table periods is considered in detail. It is shown that the period
lengths of the system of elements are determined by the structure of the basic
representation of the Rumer-Fet group. The theoretical masses of the elements
of 10th and 11th periods are calculated starting from $Z=221$ to $Z=364$. The
concept of hypertwistor is introduced.
