Evidence for an extraterrestrial flux of high-energy neutrinos has now been found in multiple searches with the IceCube detector. The first solid evidence was provided by a search for neutrino events with deposited energies greater than or similar to 30 TeV and interaction vertices inside the instrumented volume. Recent analyses suggest that the extraterrestrial flux extends to lower energies and is also visible with throughgoing, nu(mu)-induced tracks from the Northern Hemisphere. Here, we combine the results from six different IceCube searches for astrophysical neutrinos in a maximum-likelihood analysis. The combined event sample features high-statistics samples of shower-like and track-like events. The data are fit in up to three observables: energy, zenith angle, and event topology. Assuming the astrophysical neutrino flux to be isotropic and to consist of equal flavors at Earth, the all-flavor spectrum with neutrino energies between 25 TeV and 2.8 PeV is well described by an unbroken power law with best-fit spectral index -2.50 +/- 0.09 and a flux at 100 TeV of (6.7(-1.2)(+1.1)) x 10(-18) GeV-1 s(-1) sr(-1) cm(-2). Under the same assumptions, an unbroken power law with index -2 is disfavored with a significance of 3.8 sigma (p = 0.0066%) with respect to the best fit. This significance is reduced to 2.1 sigma (p = 1.7%) if instead we compare the best fit to a spectrum with index -2 that has an exponential cut-off at high energies. Allowing the electron-neutrino flux to deviate from the other two flavors, we find a nu(e) fraction of 0.18 +/- 0.11 at Earth. The sole production of electron neutrinos, which would be characteristic of neutron-decay-dominated sources, is rejected with a significance of 3.6 sigma ( p = 0.014%).