Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C5×C25 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5×D25 |
| commutator, Fitting, cosocle | C2×C5×C25 | C22 |
| Frattini | C2×C5 | D52 |
| socle | C2×C52 | C2×D5 |
| hypocenter | C5×C25 | D4 |
| LCS | G ▷ C2×C5×C25 ▷ C5×C25 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 200 |
Polycyclic presentation:
2^3,5^3|d^5=f,b^a=bc,e^a=e^4,d^b=d^4f^4,f^b=f^4109 conjugacy classes — view the full character table.
| maximal subgroup of | #14522389, #14522394, #14522396, … |
| maximal normal subgroup of | #14522389, #14522394, #14522396, … |
| maximal quotient of | #14522120, #14522122, #14522124, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 40);