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×#125 |
| 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 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 800 |
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)(28,32)(30,33)(34,35,37,39)(36,38,40,41);(26,27,29,31)(28,30,32,33)(34,35)(36,38)(37,39)(40,41);(26,28)(27,30)(29,32)(31,33)(34,36,37,40)(35,38,39,41)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 186);