Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C52 · Schur multiplier (H2) = C22×C5 |
| direct product | C52×#430 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C52 | C2×D4 |
| commutator | C4 | C22×C4×C52 |
| Frattini, cosocle | C2×C4 | C23×C52 |
| socle | C2×C52 | C2×C4⋊C4 |
| prime radical | C23×C52 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C52 ◁ C22×C4×C52 ◁ G | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3,4,5)(11,12,15,18,16,19,14,24)(13,20,23,25,21,26,22,17);(6,7,8,9,10)(11,13,14,22,16,21,15,23)(12,17,18,20,19,25,24,26);(13,21)(17,25)(20,26)(22,23)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5519);