Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C52 · Schur multiplier (H2) = C2×C5 |
| direct product | C52×#358 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C52 | D8 |
| commutator | C8 | C2×C4×C52 |
| Frattini, cosocle | C2×C8 | C22×C52 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ C2×C4×C52 ◁ C2×C52×C8 ◁ G | |
232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1600 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3,4,5)(11,12,13,14)(16,20)(17,23)(18,25)(21,30)(22,29)(24,27)(26,28);(6,7,8,9,10)(15,16,19,22)(17,24,26,30)(18,27,25,21)(20,28,29,23)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1991);