Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C52 · Schur multiplier (H2) = C2×C5 |
| direct product | C52×#362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C52 | #102 |
| 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 | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 2 generators on 42 points:
(1,2,3,4,5)(11,12,15,19,16,21,31,20)(13,22,24,35,25,39,30,38)(14,26,28,18,29,36,42,37)(17,27,32,23,34,40,33,41);(6,7,8,9,10)(11,13,16,25)(12,17,21,34)(14,27,29,40)(15,30,31,24)(18,35,37,38)(19,33,20,32)(22,26,39,36)(23,28,41,42)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1995);