Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C25 · Schur multiplier (H2) = C2 |
| direct product | C25×C4⋊C8 |
| unambiguous semidirect product | C4⋊(C8×C25) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C25 | C22 |
| commutator | C2 | C2×C8×C25 |
| Frattini, cosocle | C2×C4×C5 | C22×C5 |
| socle, prime radical | C22×C5 | C2×C4×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C25 ◁ G | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 2 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)(27,29)(30,31,32,34,33,35,36,37);(26,27,28,29)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 55);