Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C52 · Schur multiplier (H2) = C2×C5 |
| direct product | C52×#126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C52 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C52 |
| prime radical | C23×C52 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3,4,5)(12,16)(13,20)(18,25)(19,23)(21,24)(22,26);(6,7,8,9,10)(11,12,17,23)(13,18,24,25)(14,16,15,19)(20,26,21,22);(11,13,14,21)(12,18,16,22)(15,20,17,24)(19,26,23,25)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 945);