Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12806 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #410 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | #1799 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×C8 ◁ C2×C5×#131 ◁ G | |
420 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)(10,23,24,13)(11,15)(12,18)(14,22)(17,31)(19,29,33,30)(21,28,36,35)(25,37,26,34)(27,32);(6,8,9,18,11,17,22,32)(7,12,14,16,15,27,20,31)(10,19,21,34,24,33,36,37)(13,26,28,30,23,25,35,29);(7,13)(8,17)(12,25)(14,28)(15,23)(16,29)(18,32)(19,33)(20,35)(26,27)(30,31)(34,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5794);