Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C52 · Schur multiplier (H2) = C1 |
| direct product | Q16×5−1+2 |
| unambiguous semidirect product | (Q16×C25)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C52×D4 |
| commutator, Frattini, cosocle | C4×C5 | C22×C52 |
| prime radical | C2×C52 | C5×D4 |
| LCS | G ▷ C4×C5 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C4×5−1+2 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,20,24,31,19,22,26,33,38,23,29,35,32,39,30,28,25,34,40,37,21,27,36,41);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,21,28,29,22)(20,25,26,27,35)(24,32,36,33,34)(31,38,39,40,41)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 233);