Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12768 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1790 |
| commutator | C2×D4 | C22×C4×C5 |
| Frattini, cosocle | C22×D4 | C23×C5 |
| prime radical | C5×#3545 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×D4 ◁ G | |
778 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)(9,20,22,13)(10,23,24,14)(11,15)(12,18,26,33)(17,29,32,36)(19,30)(21,27)(25,31)(28,35)(34,37);(6,8,11,19)(7,12,15,26)(9,17,22,32)(10,18,24,33)(13,25,20,34)(14,16,23,30)(21,31,35,37)(27,36,28,29);(6,9)(7,13)(8,17)(10,21)(11,22)(12,25)(14,28)(15,20)(16,29)(18,31)(19,32)(23,27)(24,35)(26,34)(30,36)(33,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5756);