Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21785 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2308 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,18,22,13)(9,15,10,24)(11,17)(12,16)(14,23,32,25)(19,35)(20,29)(21,34,26,30)(27,33)(28,36)(31,37);(6,8)(7,13)(9,19)(10,20)(11,26)(12,22)(14,31)(15,29)(16,18)(17,34)(21,27)(23,37)(24,35)(25,36)(28,32)(30,33);(6,9,12,10)(7,14,16,32)(8,19,22,20)(11,25,27,23)(13,28,18,31)(15,17,24,33)(21,37,26,36)(29,30,35,34);(6,10,12,9)(7,15,16,24)(8,20,22,19)(11,25,27,23)(13,29,18,35)(14,17,32,33)(21,37,26,36)(28,30,31,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14773);