  We use the conformal bootstrap to perform a precision study of 3d maximally
supersymmetric ($\mathcal{N}=8$) SCFTs that describe the IR physics on $N$
coincident M2-branes placed either in flat space or at a $\C^4/\Z_2$
singularity. First, using the explicit Lagrangians of ABJ(M)
\cite{Aharony:2008ug,Aharony:2008gk} and recent supersymmetric localization
results, we calculate certain half and quarter-BPS OPE coefficients, both
exactly at small $N$, and approximately in a large $N$ expansion that we
perform to all orders in $1/N$. Comparing these values with the numerical
bootstrap bounds leads us to conjecture that some of these theories obey an OPE
coefficient minimization principle. We then use this conjecture as well as the
extremal functional method to reconstruct the first few low-lying scaling
dimensions and OPE coefficients for both protected and unprotected multiplets
that appear in the OPE of two stress tensor multiplets for all values of $N$.
We also calculate the half and quarter-BPS operator OPE coefficients in the
$SU(2)_k \times SU(2)_{-k}$ BLG theory for all values of the Chern-Simons
coupling $k$, and show that generically they do not obey the same OPE
coefficient minimization principle.
