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We consider the free nilpotent Lie algebra $L$ with 2 generators, of step 4,and the corresponding connected simply connected Lie group $G$. We study theleft-invariant sub-Riemannian structure on $G$ defined by the generators of $L$as an orthonormal frame.  We compute two vector field models of $L$ by polynomial vector fields in$R^8$, and find an infinitesimal symmetry of the sub-Riemannian structure.Further, we compute explicitly the product rule in $G$, the right-invariantframe on $G$, linear on fibers Hamiltonians corresponding to the left-invariantand right-invariant frames on $G$, Casimir functions and co-adjoint orbits on$L^*$.  Via Pontryagin maximum principle, we describe abnormal extremals and derive aHamiltonian system $\dot \lambda = \vec{H}(\lambda)$, $\lambda \in T^*G$, fornormal extremals.  We compute 10 independent integrals of $\vec{H}$, of which only 7 are ininvolution. After reduction by 4 Casimir functions, the vertical subsystem of$\vec{H}$ on $L^*$ shows numerically a chaotic dynamics, which leads to aconjecture on non-integrability of $\vec{H}$ in the Liouville sense.

Sub-Riemannian geodesics on the free Carnot group with the growth vector (2,3,5,8)