Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C25 · Schur multiplier (H2) = C2 |
| direct product | C25×D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C25 | D8 |
| commutator | C8 | C22×C25 |
| Frattini, cosocle | C5×C8 | C22×C5 |
| socle | C2×C5 | C5×D8 |
| prime radical | C5×D16 | C5 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C25 ◁ C4×C25 ◁ C8×C25 ◁ G | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 61);