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×#441 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C52 | C2×D4 |
| commutator | C4 | C22×C4×C52 |
| Frattini, cosocle | C2×C4 | C23×C52 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ C42×C52 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1600 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3,4,5)(11,12,14,13)(15,16)(17,23)(18,26)(19,21)(20,24)(22,28)(25,30)(27,29);(6,7,8,9,10)(15,17,19,25)(16,20,21,29)(18,27,28,24)(22,30,26,23);(11,13,14,12)(16,21)(20,29)(22,26)(23,30)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5530);