Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C8 · Schur multiplier (H2) = C1 |
| direct product | C8×5−1+2 |
| unambiguous semidirect product | (C8×C25)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C52 |
| commutator | C5 | C52×C8 |
| Frattini, cosocle | C4×C5 | C2×C52 |
| socle | C2×C5 | C4×C52 |
| prime radical | C2×C52 | C4×C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8)(9,10,12,16,23,11,14,18,25,30,15,21,27,24,31,22,20,17,26,32,29,13,19,28,33);(10,13,20,21,14)(12,17,18,19,27)(16,24,28,25,26)(23,30,31,32,33)232 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 23);