The recent observation that pressure could suppress antiferromagnetic (AFM) order in quasi-one-dimensional $A{\mathrm{Mn}}_6{\mathrm{Bi}}_5$ Mn-cluster chain materials ($A=\mathrm{Na}$, K, Rb, and Cs) and lead to a superconducting dome offers an alternative Mn-based class of materials with which to study unconventional superconductivity. Using neutron diffraction, we elucidate the exact nature of the previously unknown AFM ground state of ${\mathrm{KMn}}_6{\mathrm{Bi}}_5$ and report finding transverse incommensurate spin density waves (SDWs) for the Mn atoms with a propagating direction along the chains. The SDWs have distinct refined amplitudes of $\sim 2.46{\mu }_{\mathrm{B}}$ for the Mn atoms in the pentagons and $\sim 0.29{\mu }_{\mathrm{B}}$ with a large standard deviation for Mn atoms at the center between the pentagons. AFM coupling dominates both the nearest-neighbor Mn-Mn interactions within the pentagon and next-nearest-neighbor Mn-Mn interactions out of the pentagon (along the propagating wave). The SDWs exhibit both local and itinerant characteristics potentially due to cooperative interactions between local magnetic exchange and conduction electrons. Single crystal x-ray diffraction below the AFM transition revealed satellite peaks originating from charge density waves along the chain direction with a $\mathbf{q}$ vector twice as large as that of the SDW, pointing to a strong real space coupling between them. Additionally, we report a significant magnetoelastic effect during the AFM transition, especially along the chain direction, observed in temperature-dependent x-ray powder diffraction. Our work not only reveals fascinating intertwined spin, charge, and lattice orders in one-dimensional ${\mathrm{KMn}}_6{\mathrm{Bi}}_5$, but also provides an essential piece of information on its magnetic structure to understand the mechanism of superconductivity in this Mn-based family.

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• Charge density waves
• Magnetic order
• Spin density waves
• 1-dimensional systems
• Single crystal materials
• Superconductors
• Neutron diffraction
• X-ray diffraction