Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| isoclinism | isoclinic class of C5⋊D125 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C5⋊D125 |
| unambiguous semidirect product | C125⋊D5 |
| Fitting, derived, Sylow | C5×C125 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C5×C125 | C2 |
| Frattini | C25 | C5⋊D5 |
| socle | C52 | D25 |
| LCS | G ▷ C5×C125 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 625 |
Polycyclic presentation:
2,5^4|b^5=d,d^5=e,b^a=b^4d^4e^4,c^a=c^4,d^a=d^4e^4,e^a=e^4314 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 25);