Suppose I have a noncyclic group defined as follows (I'm probably butchering the math here because I don't completely understand):
$G$ consisting of elements which are the set of cosets under multiplication by 2 $\bmod 2^N-1$, excluding elements having a common factor with $2^N-1$, using multiplication as the operation.
For example, if $N=8$ then the cosets are
$$\begin{align} \overline{1} &= \{1,2,4,8,16,32,64,128\} \cr \overline{7} &= \{7,14,28,56,112,224,193,131\} \cr \overline{11} &= \{11,22,44,88,176,97,194,133\} \cr \overline{13} &= \{13,26,52,104,208,161,67,134\} \cr \overline{19} &= \{19,38,76,152,49,98,196,137\} \cr \overline{23} &= \{23,46,92,184,113,226,197,139\} \cr \overline{29} &= \{29,58,116,232,209,163,71,142\} \cr \overline{31} &= \{31,62,124,248,241,227,199,143\} \cr \overline{37} &= \{37,74,148,41,82,164,73,146\} \cr \overline{43} &= \{43,86,172,89,178,101,202,149\} \cr \overline{47} &= \{47,94,188,121,242,229,203,151\} \cr \overline{53} &= \{53,106,212,169,83,166,77,154\} \cr \overline{59} &= \{59,118,236,217,179,103,206,157\} \cr \overline{61} &= \{61,122,244,233,211,167,79,158\} \cr \overline{91} &= \{91,182,109,218,181,107,214,173\} \cr \overline{127} &= \{127,254,253,251,247,239,223,191\} \end{align}$$
For example, $\overline{7} \times \overline{7} = \overline{19}$ since 49 is in the coset with 19.
This doesn't have a single generator.
Is there a systematic way to find generators that span the entire group? I can figure out that if I use $\alpha = \overline{7}$ and $\beta = \overline{13}$ then any element can be written as $\alpha^i\beta^j$ for some $i,j$. But I can't seem to generalize this.
And is there a way to find out the period of an element $\alpha$ in the group without exhaustively calculating $\alpha^i$ until I reach $\overline{1}$ again?
(For example, if $N=32$ then I don't know how to find the period of $\overline{7}$ without trying it thousands or even millions of times.)
I don't have much experience with noncyclic groups and not sure how to approach.
