Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C25 · Schur multiplier (H2) = C2 |
| direct product | C25×#126 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C25 | C23 |
| commutator | C22 | C23×C25 |
| Frattini, socle, cosocle | C22×C5 | C23×C5 |
| prime radical | C23×C5 | C22×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C25 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 3 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)(27,31)(28,35)(33,40)(34,38)(36,39)(37,41);(26,27,32,38)(28,33,39,40)(29,31,30,34)(35,41,36,37);(26,28,29,36)(27,33,31,37)(30,35,32,39)(34,41,38,40)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 187);