Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C52 · Schur multiplier (H2) = C4×C5 |
| direct product | C52×#490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C52 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C52 |
| prime radical | C52×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ C42×C52 ◁ G | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3,4,5)(11,12)(13,16)(14,15)(17,18)(20,25)(21,26)(22,23);(6,7,8,9,10)(12,15)(16,18)(19,20,22,25,24,26,23,21);(11,13,14,17)(12,16,15,18)(19,21,22,20,24,25,23,26)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5579);