Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽5)⋊C5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C52 · Schur multiplier (H2) = C5 |
| direct product | C2×#10597833 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C52 | C2×C52 |
| prime radical | C2×H(3,𝔽5) | C5 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C52 ◁ G | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 250 |
98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 10);