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×C4 |
| direct product | C25×#124 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C25 | C23 |
| commutator | C22 | C23×C25 |
| Frattini, socle, cosocle | C22×C5 | C23×C5 |
| prime radical | C2×C5×D4 | C2×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C25 ◁ G | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 3 generators on 37 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)(27,28)(32,36)(34,37);(26,27,29,28)(30,31,33,35)(32,34,36,37);(26,28,29,27)(30,32)(31,34)(33,36)(35,37)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 185);