Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C25 · Schur multiplier (H2) = C1 |
| direct product | C2×C125⋊C5 |
| unambiguous semidirect product | (C2×C125)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C25 | C52 |
| commutator | C5 | C2×C5×C25 |
| Frattini, cosocle | C25 | C2×C52 |
| socle | C2×C5 | C5×C25 |
| prime radical | C2×C52 | C25 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C2×C25 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 250 |
290 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | C2×D125⋊C5, C2×#72229806 |
| maximal subgroup of | C2×D125⋊C5 |
| maximal normal subgroup of | C2×D125⋊C5 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 6);