Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C53 |
No aliases for this group.
| Fitting | C53×C8 ⋊ C2 |
| Sylow | C53 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊(C5⋊(C5⋊D5)) |
| commutator, cosocle | C4×C53 | C22 |
| Frattini | C4 | C2×C5⋊(C5⋊D5) |
| Fitting | C53×C8 | C2 |
| socle | C2×C53 | D4 |
| hypercenter | C8 | C5⋊(C5⋊D5) |
| hypocenter | C53 | D8 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
704 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1000 |
Polycyclic presentation:
2^4,5^3|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^4,g^a=g^4503 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 604);