Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 16000 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2×C5 |
| abelian | C2×C4×C5×C25 |
| multiplicative | (ℤ/1111)ˣ, (ℤ/1375)ˣ, (ℤ/2222)ˣ, (ℤ/2750)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C5 | C22×C52 |
| socle, prime radical | C22×C52 | C2×C5 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 1000 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31);(5,6)(32,33,34,35,36)1000 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 77);