Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33388 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33392 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#102 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#576 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,10,15,11,17,25,16)(8,13,20,30,18,28,33,31)(9,21,23,14,24,12,22,27)(19,34,36,29,32,26,35,37);(7,12)(9,22)(10,25)(13,26)(14,15)(16,27)(17,21)(19,35)(20,33)(23,24)(28,34)(29,30)(31,37)(32,36);(6,8)(7,13)(9,19)(10,20)(11,18)(12,26)(14,29)(15,30)(16,31)(17,28)(21,34)(22,35)(23,36)(24,32)(25,33)(27,37);(8,18)(13,28)(19,32)(20,33)(26,34)(29,37)(30,31)(35,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26380);