Finite Group DatabasebetaFinite Group Databasebeta| order | 15625 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C53 · Schur multiplier (H2) = C53 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C5×C25 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C53 | C53 |
| socle | C52 | C5×H(3,𝔽5) |
| prime radical | #15691060 | C5 |
| LCS | G ▷ C53 ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C5×C25 ◁ C52×C25 ◁ G | |
| LECS | G ▷ C53 ▷ C52 ▷ C1 | |
513 conjugacy classes of subgroups.
Polycyclic presentation:
5^6|c^5=e^4,b^a=bd,d^a=de,d^b=df745 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15625, 129);