Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| isoclinism | isoclinic class of C5⋊((C5⋊D5)⋊C5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C52 |
| unambiguous semidirect product | C5⋊((C5⋊D5)⋊C5) |
| Fitting, Sylow | C5×H(3,𝔽5) ⋊ C2 |
| derived | C53 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C53 | C2×C5 |
| Frattini | C5 | C5×C5⋊D5 |
| Fitting | C5×H(3,𝔽5) | C2 |
| socle | C52 | C5×D5 |
| LCS | G ▷ C53 | |
| UCS | C1 | |
122 conjugacy classes of subgroups.
| transitive degree | μt(G) = 125 |
Polycyclic presentation:
2,5^4|b^a=be^4,c^a=c^4,d^a=d^4,e^a=e^4,c^b=ce80 conjugacy classes — view the full character table.
| maximal subgroup of | #72229816 |
| maximal normal subgroup of | #72229816 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 44);