Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11138 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #396 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1439 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,20,23,13)(10,24)(11,15)(12,18)(14,26)(17,33,21,31)(19,32)(22,29,25,30)(27,35,37,34)(28,36);(6,8,10,18,11,19,26,36)(7,12,14,16,15,28,24,32)(9,21,22,35,23,17,25,34)(13,27,29,33,20,37,30,31);(6,9)(7,13)(8,17)(10,25)(11,23)(12,27)(14,30)(15,20)(16,31)(18,35)(19,21)(22,26)(24,29)(28,37)(32,33)(34,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4126);