Finite Group DatabasebetaFinite Group Databasebeta| order | 15625 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C625 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C25⋊C625 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C5×C125 | C52 |
| commutator | C5 | C5×C625 |
| socle, prime radical | C52 | C5×C125 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C125 ◁ G | |
| LECS | G ▷ C5×C125 ▷ C25 ▷ C5 ▷ C1 | |
55 conjugacy classes of subgroups.
Polycyclic presentation:
5^6|a^5=c^4,b^5=d,d^5=e,e^5=f,b^a=bc3625 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15625, 21);