It's definitely a Dieffenbachia.
Your variety is not pictured, but example species from this genus at:
http://www.plantcare.com/encyclopedia/dumb-cane-387.aspx
(and plant care, too!)

Propagation:
Stem cuttings, suckers, air-layering at anytime.
It is best to propagate any type of cuttings or suckers in a mixture of moist peat and perlite.
Cover the pot and plant with a plastic bag secured by a rubber band to prevent the moisture from escaping.
Place in indirect sunlight or under a fluorescent light.
Repot in its regular mix after it has been growing for a while.
...but be careful not to damage new roots.

BEST is:
To air-layer a plant.
(This method is good for most plants that get long and leggy.)
Take a sharp knife and cut half way through (No! I'd go just to the pith. Approx 1/4 of diameter) the stem about a foot down from the top (actually, anywhere beneath the lowest leaves).
Put a toothpick in the wound sideways (to keep the wound open), wrap wet sphagnum moss and a plastic bag (or food wrap) around the wound secured by either rubber bands or tape (or twist-ties)(You may want to cover with foil to prevent drying-out.
Either way, position wrap so you can access from the top-side. This allows you to mist the moss to keep it moist, when condensation is not present inside the plastic wrap)
When roots form, cut off below the roots and plant.
...but be careful not to damage new roots.
Either method:
If there are many leaves, trim off the lower, older leaves to balance transpiration of moisture. Too much leaf area will cause leaves to wilt; because there is no roots to draw moisture and transpiration and photosynthesis will be out of balance.