The structure of the Froebel gifts can be explored in terms of the isometries that leave them invariant. This classification is similar to the classifications employed for the partition , say, of the Euclidean group, E, consisting of all isometries, to the ten groups Eij, or the partition of the three dimensional point group , E03, consisting of all rotations, reflections and rotor reflections about a point 0, in the seven polyhedral groups and the seven infinite sets of point symmetry ; the difference is that while those classifications are built upon classes of equivalent isometries in the Euclidean space, i.e. all reflections or all n-fold rotations are of the same type respectively, this classification distinguishes between these symmetries, so far considered typologically the same, in respects to the effect that they impose on each individual spatial structure . In these corresponding spaces, neither all reflections are the same, nor all two-fold rotations are of the same type. In the cube, for example, a reflection in a face plane interchanges all three pairs of faces while a reflection in an edge plane interchanges two faces and leaves two faces still. Both isometries permute the faces in a different way and thus they are classified as different. Informally, two isometries are equivalent or conjugate when they impose the same kind of rearrangement in the structure. Formally, given elements x, y of a group G, xis conjugate toy if gxg-1 = y for some g E G. The set of all conjugates of x, {gxg-1 19 E G} is called the conjugacy class of x in G and all the elements in the same class have the same order.

Within this context the forty-eight elements of the symmetry group of the cube naturally split in ten conjugacy classes (Yale, 1988). The conjugacy classes of the symmetries of the other three Froebel blocks are subsets of these ten classes. The sixteen elements of the pillar are partitioned in eight classes, the eight elements of the oblong in four classes and the four elements of the half-cube in three classes. The explicit descript ions of these types of symmetries and their order for the four Froebel blocks are given in Table 3. These conjugate classes of symmetries combine to create conjugate groups within the symmetry groups of each individual solid. The conjugate subgroups show essentially the different types of symmetries within each block . There are thirty-three conjugate classes in the octahedral group and it can easily deduce that the symmetry groups of the pillar, the oblong and the half-cube can be further partitioned in twenty-three , eight and four conjugate classes respectively. The symmetry groups can be even further scrutinized in terms of all the distinct subgroups that are formed out of all possible combinations of the elements of the group.

The structure of the cube, the octahedral group, has ninety-eight subgroups which all together form the thirty-three conjugacy classes of the cube. All the subgroups of the other Froebel gifts are contained within these ninety-eight groups. It can be easily checked that the symmetry group of the pillar contains thirty-five subgroups, the symmetry group of the oblong contains sixteen subgroups , and the symmetry group of the half-cube contains four subgroups . The complete number of all conjugate subgroups for the cube is given here for a spatial relations between a cube and an oblong.