Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11094 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #392 |
| 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 | |
868 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)(6,7,11,15,12,16,10,25)(8,13,20,28,19,29,18,34)(9,21,24,14,23,31,26,30)(17,32,33,27,22,36,35,37);(6,8)(7,13)(9,22,23,17)(10,18)(11,20)(12,19)(14,27,30,37)(15,28)(16,29)(21,32,31,36)(24,35,26,33)(25,34);(6,9)(7,14)(8,17)(10,26)(11,24)(12,23)(13,27)(15,31)(16,30)(18,35)(19,22)(20,33)(21,25)(28,36)(29,37)(32,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4082);