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×C4 |
| direct product | C25×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C25 | C23 |
| commutator | C22 | C23×C25 |
| Frattini, socle, cosocle | C22×C5 | C23×C5 |
| prime radical | C5×#127 | C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C25 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 3 generators on 33 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)(31,32);(30,31,33,32);(26,27,28,29)(30,32,33,31)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 188);