Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11277 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #387 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1442 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
592 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)(8,16,19,12)(9,20)(10,15)(11,14)(13,22)(17,26,34,32)(18,33,36,27)(21,28,35,37)(23,31)(24,29)(25,30);(6,8)(7,12)(9,21,22,35)(10,18)(11,19)(13,28,20,37)(14,16)(15,27)(17,23,34,24)(25,36)(26,29,32,31)(30,33);(6,9)(7,13,14,20)(8,17)(10,24)(11,22)(12,26,16,32)(15,31,30,29)(18,35)(19,34)(21,36)(23,25)(27,37,33,28)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4265);