Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C25 · Schur multiplier (H2) = C5 |
| direct product | C5×SD16×C25 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C25 | D4 |
| commutator | C4 | C22×C5×C25 |
| Frattini, cosocle | C4×C5 | C22×C52 |
| socle | C2×C52 | C5×D4 |
| prime radical | C52×D4 | C2×C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C25 ◁ C4×C5×C25 ◁ G | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 1000 |
A minimal faithful permutation representation — 2 generators on 38 points:
(1,2,3,4,5)(31,32,33,34,35,36,37,38);(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(32,34)(33,37)(36,38)875 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 228);