Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C25 · Schur multiplier (H2) = C22 |
| direct product | C25×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C25 | C22 |
| commutator | C2 | C22×C4×C25 |
| Frattini, cosocle | C4×C5 | C23×C5 |
| socle | C2×C5 | C22×C4×C5 |
| prime radical | C5×#41 | C2×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C25 ◁ G | |
93 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)(26,27,29,32,30,33,36,39)(28,31,35,38,34,37,40,41);(26,28)(27,31)(29,35)(30,34)(32,38)(33,37)(36,40)(39,41);(28,34)(31,37)(35,40)(38,41)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 192);