Dekabutsu is a huge Dai-Gun-far larger than Dai-Gurren-which functions as the capital city of Teppelin under Lordgenome's rule. After it is destroyed in the battle of Teppelin, it is rebuilt by the new government and made into Parliament Tower.
Dekabutsu is extremely bulky, and has a very uncomplicated humanoid shape with tiny legs and massive arms. One arm has a three-fingered hand, while the other ends in a giant hammer. It uses its arms in combat, and can also deploy a vast amount of Gunmen. The tower on top of Dekabutsu's head is its docking point for Lazengann. It can be assumed that, after Lazengann docked with Dekabutsu, the tower was constructed around it.
The tower also provides a unique interface, allowing Lordgenome to control the entire Gunman without needing to be inside Lazengann. The Dekabutsu itself seems to project energy to all allied gunman, as Beastmen-piloted Gunmen are usually unable to function during the night (which they did during the battle of Teppelin) and the fact that all Beastmen Gunmen shut down right after Lordgenome's defeat.
Before Beastman War Arc
It is shown in Parallel Works episode 8, and the beginning of the first movie (Gurren Lagann The Movie: Childhood's End), that the Dekabutsu was used by Lordgenome during his time as a Spiral warrior to fight against the Anti-Spirals. Specifically, Lazengann is shown to dock with Dekabutsu in the same way that Gurren-Lagann docks with the Arc-Gurren, and it acts as the spiral engine of Cathedral Terra.
After Lordgenome learns of the Spiral Nemesis-shown to him by the Anti-Spiral-he uses Lazengann to take command of Dekabutsu, transforming Cathedral Terra into its massive humanoid form and obliterating his former allies. After this, the Dekabutsu lands back on earth and begins the suppression of humans by unleashing legions of Beastman-powered Gunmen as its footsoldiers. Over time, the Gunman is concealed behind hundreds of towers, becoming Teppelin Castle.
During Beastman War ArcThe Dekabutsu would not be reactivated until a large rebel force, led by Team Dai-Gurren, arrived at the capital. After the defeat of Generals Guame and Cytomander, Teppelin sheds its surrounding towers to reveal that it is, in fact, a massive Gunman. Spotting Lordgenome in his throne room, Simon charges the tower on the head of Dekabutsu, but is stopped by one of its fingers, which is tough enough to withstand Gurren-Lagann's Giga Drill before flicking it away. Dekabutsu then releases an armada of Gunmen and Dai-Guns to fight against Team Dai-Gurren.
Undaunted, Simon and Gurren-Lagann return, with Dai-Gurren backing them up, and ram themselves into the Dekabutsu's hammer arm, puncturing it. Dai-Gurren unloads all of its cannons within the weapon, creating a hole straight to Lordgenome's throne room, and launches Gurren-Lagann into it, allowing Simon to fight Lordgenome. Meanwhile, Dai-Gurren then blows up its own hull, destroying Dekabutsu's hammer arm while ejecting to safety with its crew.
After the Spiral King's defeat, the Dekabutsu automatically shuts down and collapses.
Before Anti-Spiral War Arc
During the seven years of peace, Dekabutsu was converted into a capital building called Parliament Tower, where Team Dai-Gurren governed the people from. Simon himself had an office at the top.
During Anti-Spiral War Arc
The Parliament Tower becomes the target of attack by the Anti-Spirals as they attempt to raze it in order to reach the Arc-Gurren, whose spiral engine attracted them.