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×#11361 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #387 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1445 |
| 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)(6,7,11,15,12,17,10,16)(8,18,21,29,22,33,20,13)(9,23,25,14,27,31,26,32)(19,35,24,34,37,28,36,30);(6,8)(7,13,17,29)(9,24)(10,20)(11,21)(12,22)(14,30,32,34)(15,18,16,33)(19,26)(23,28,31,35)(25,37)(27,36);(6,9)(7,14,17,32)(8,19)(10,25)(11,26)(12,27)(13,28,29,35)(15,23,16,31)(18,34,33,30)(20,24)(21,36)(22,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4349);