Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C25 · Schur multiplier (H2) = C4 |
| direct product | C25×#491 |
| 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 | |
135 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)(28,31)(29,30)(32,33)(34,35,40,45)(36,42,47,48)(37,41,46,49)(38,44,39,43);(27,30)(31,33)(34,36,38,37,40,47,39,46)(35,41,43,42,45,49,44,48);(26,28,29,32)(27,31,30,33)(34,37,38,47,40,46,39,36)(35,42,43,49,45,48,44,41)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 992);