Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#29470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,10)(11,13)(14,15)(16,20)(17,19)(18,21)(22,23,27,32,28,33,26,31)(24,29,34,30,25,35,36,37);(7,10)(11,12)(14,16)(15,18)(17,21)(19,20)(22,24)(23,29)(25,28)(26,36)(27,34)(30,32)(31,37)(33,35);(16,21)(18,20)(22,25,28,24)(23,30,33,37)(26,36,27,34)(29,31,35,32);(6,8,9,13)(7,11,10,12)(15,19)(18,20)(22,26,28,27)(23,31,33,32)(24,34,25,36)(29,30,35,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22458);