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×#11131 |
| 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 | |
852 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)(6,7,11,15,12,17,10,16)(8,18,22,28,21,29,20,13)(9,14,24,32,25,33,26,31)(19,34,23,27,35,30,36,37);(6,8,10,20,12,21,11,22)(7,13,15,18,17,28,16,29)(9,23,24,35,25,36,26,19)(14,30,31,27,33,34,32,37);(6,9)(7,14)(8,19)(10,26)(11,24)(12,25)(13,27)(15,32)(16,31)(17,33)(18,30)(20,36)(21,35)(22,23)(28,37)(29,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4119);