Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C25 · Schur multiplier (H2) = C4 |
| direct product | C25×#488 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C25 | C2×D4 |
| commutator | C2×C4 | C23×C25 |
| Frattini, cosocle | C2×C4×C5 | C23×C5 |
| socle | C22×C5 | C2×C5×D4 |
| prime radical | C2×C5×D4 | C22×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C25 ◁ C42×C25 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 800 |
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)(28,31)(29,30)(32,33)(34,35)(36,44)(37,40)(38,48)(39,43)(41,46)(42,45)(47,49);(27,30)(31,33)(34,36,45,47,39,46,38,37)(35,40,48,41,43,49,42,44);(26,28,29,32)(27,31,30,33)(34,37,38,46,39,47,45,36)(35,41,42,40,43,44,48,49)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 989);