Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| isoclinism | isoclinic class of C5⋊(C5⋊(C5⋊D5)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C56 |
| unambiguous semidirect product | C5⋊(C5⋊(C5⋊D5)) |
| Fitting, socle, derived, Sylow | C54 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C54 | C2 |
| LCS | G ▷ C54 | |
| UCS | C1 | |
2240 conjugacy classes of subgroups.
| transitive degree | μt(G) = 625 |
Polycyclic presentation:
2,5^4|b^a=b^4,c^a=c^4,d^a=d^4,e^a=e^4314 conjugacy classes — view the full character table.
| maximal subgroup of | C5⋊D52, D5×C5⋊(C5⋊D5), #72259182, … |
| maximal normal subgroup of | C5⋊D52, D5×C5⋊(C5⋊D5), #72259182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 54);