Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22×C53 |
| direct product | C53×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C53 |
| socle | C2×C53 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ G | |
1280 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)(18,21)(19,20)(22,23);(6,7,8,9,10)(17,20)(21,23);(11,12,13,14,15)(16,18,19,22)(17,21,20,23)1250 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 895);