Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C52×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C8×5−1+2 |
| unambiguous semidirect product | (C2×C8×C25)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C52 |
| commutator | C5 | C2×C52×C8 |
| Frattini, cosocle | C4×C5 | C22×C52 |
| socle | C22×C5 | C4×C52 |
| prime radical | C22×C52 | C4×C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2)(11,12,14,18,25,13,16,20,27,32,17,23,29,26,33,24,22,19,28,34,31,15,21,30,35);(3,4,5,6,7,8,9,10)(12,15,22,23,16)(14,19,20,21,29)(18,26,30,27,28)(25,32,33,34,35)464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 221);