Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#92562 |
| Fitting | C4×C53 ⋊ C4 |
| Sylow | C53 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#92562 |
| commutator, socle | C2×C53 | C2×C4 |
| Fitting | C4×C53 | C4 |
| prime radical, cosocle | C2×C5⋊(C5⋊D5) | C22 |
| hypercenter | C4 | #92562 |
| hypocenter | C53 | C4⋊C4 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 | |
832 conjugacy classes of subgroups.
| transitive degree | μt(G) = 500 |
Polycyclic presentation:
2^4,5^3|a^2=c,b^2=d,b^a=bd,e^a=e^2,f^a=f^2,g^a=g^2,e^c=e^4,f^c=f^4,g^c=g^4134 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 775);