Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C25 · Schur multiplier (H2) = C22 |
| direct product | C25×C4⋊Q8 |
| unambiguous semidirect product | C4⋊(Q8×C25) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C25 | C23 |
| commutator | C22 | C23×C25 |
| Frattini, socle, prime radical, cosocle | C22×C5 | C23×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C25 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(27,29)(30,31,34,37)(32,36,33,35);(30,32,34,33)(31,35,37,36);(26,27,28,29)(30,33,34,32)(31,36,37,35)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 189);