Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 11904000 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2×C53 |
| abelian | C2×C4×C53 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C22×C53 |
| socle, prime radical | C22×C53 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 1000 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4)(7,8,9,10,11);(5,6)(12,13,14,15,16);(17,18,19,20,21)1000 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 159);