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×#359 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C52 | D8 |
| commutator | C8 | C2×C4×C52 |
| Frattini, cosocle | C2×C8 | C22×C52 |
| socle | C2×C52 | #102 |
| prime radical | C52×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C52 ◁ C2×C4×C52 ◁ C2×C52×C8 ◁ G | |
224 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)(11,12,15,20,17,21,16,31)(13,22,24,38,25,39,32,36)(14,26,28,37,30,41,29,19)(18,27,33,40,35,42,34,23);(6,7,8,9,10)(11,13)(12,18)(14,27)(15,24)(16,32)(17,25)(19,36)(20,33)(21,35)(22,26)(23,29)(28,40)(30,42)(31,34)(37,38)(39,41)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1992);