Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C53 · Schur multiplier (H2) = C53 |
| direct product | C53×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | C22 |
| commutator | C2 | C2×C4×C53 |
| Frattini, cosocle | C4 | C22×C53 |
| socle | C2×C53 | C2×C4 |
| prime radical | C22×C53 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ G | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 1000 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5)(16,17,18,19,20,21,22,23);(6,7,8,9,10)(17,21)(19,23);(11,12,13,14,15)1250 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 621);