Finite Group DatabasebetaFinite Group Databasebeta| order | 15625 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C125 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C25 | C25⋊C25 |
| commutator | C25 | C5×C125 |
| Frattini, cosocle | C5×C125 | C52 |
| socle | C5 | C25⋊C125 |
| prime radical | C52 | C125⋊C5 |
| LCS | G ▷ C25 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C5×C125 ◁ G | |
| LECS | G ▷ C5×C125 ▷ C25 ▷ C5 ▷ C1 | |
43 conjugacy classes of subgroups.
Polycyclic presentation:
5^6|a^5=c^4f,b^5=d,c^5=f^4,d^5=e,e^5=f,b^a=bc,d^a=df^4,c^b=cf^4745 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15625, 187);