Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| isoclinism | isoclinic class of D125⋊C5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | D125⋊C5 |
| unambiguous semidirect product | (C125⋊C5)⋊C2 |
| Fitting, Sylow | C125⋊C5 ⋊ C2 |
| derived | C125 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C125 | C2×C5 |
| Frattini | C25 | C5×D5 |
| Fitting | C125⋊C5 | C2 |
| socle | C5 | C5×D25 |
| LCS | G ▷ C125 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 125 |
Polycyclic presentation:
2,5^4|b^5=d,d^5=e,b^a=b^4d^4e^4,d^a=d^4e^4,e^a=e^4,c^b=ce80 conjugacy classes — view the full character table.
| maximal subgroup of | #72229806, C2×D125⋊C5 |
| maximal normal subgroup of | #72229806, C2×D125⋊C5 |
| maximal quotient of | C2×D125⋊C5 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 26);