Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5×C25 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D5×D25 |
| commutator, cosocle | C2×C5×C25 | C23 |
| Frattini | C2×C5 | C2×D52 |
| Fitting | C4×C5×C25 | C22 |
| socle | C2×C52 | C22×D5 |
| hypercenter | C4 | D5×D25 |
| hypocenter | C5×C25 | #41 |
| LCS | G ▷ C2×C5×C25 ▷ C5×C25 | |
| UCS | C1 ◁ C2 ◁ C4 | |
144 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 400 |
Polycyclic presentation:
2^4,5^3|b^2=d,c^2=d,e^5=g,c^a=cd,e^a=e^4g^4,g^a=g^4,f^b=f^4212 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 352);