Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C52 · Schur multiplier (H2) = C5 |
| direct product | C52×#365 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C52 | #106 |
| commutator | C8 | C2×C4×C52 |
| Frattini, cosocle | C2×C8 | C22×C52 |
| prime radical | C22×C52 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C52 ◁ C2×C4×C52 ◁ C2×C52×C8 ◁ G | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4,5)(12,16,17,22)(13,19)(14,21)(18,25,24,23)(20,26);(6,7,8,9,10)(11,12,13,16,14,18,19,23,15,17,20,22,21,24,26,25)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1998);