Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C25 · Schur multiplier (H2) = C2 |
| direct product | C25×#499 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C25 | C2×D4 |
| commutator | C2×C4 | C23×C25 |
| Frattini, cosocle | C2×C4×C5 | C23×C5 |
| socle, prime radical | C22×C5 | C2×C5×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C25 ◁ C42×C25 ◁ G | |
129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1600 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,29,31)(28,32,33,30)(34,35,37,39)(36,42,43,49)(38,46,45,41)(40,48,47,44);(26,28,29,33)(27,30,31,32)(35,39)(40,47)(41,46)(42,49);(34,36,38,44,37,43,45,48)(35,40,41,42,39,47,46,49)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1000);