Finite Group DatabasebetaFinite Group Databasebeta| order | 625 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C25 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C25⋊C25 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C52 | C52 |
| commutator | C5 | C5×C25 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52 ◁ G | |
| LECS | G ▷ C52 ▷ C1 | |
25 conjugacy classes of subgroups.
| transitive degree | μt(G) = 625 |
Polycyclic presentation:
5^4|a^5=d,b^5=c,b^a=bc145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(625, 4);