Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#99 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×Q8 | C24×C5 |
| prime radical | C5×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×D4 ◁ C2×C5×2−1+4 ◁ G | |
1176 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)(6,7,11,16,12,17,8,18)(9,14,23,30,21,28,19,32)(10,15,20,27,25,31,24,13)(22,29,33,36,34,37,35,26);(6,8,12,11)(7,13,17,27)(9,19,21,23)(10,24,25,20)(14,26,28,36)(15,16,31,18)(22,35,34,33)(29,30,37,32);(6,9)(7,14)(8,19)(10,22)(11,23)(12,21)(13,26)(15,29)(16,30)(17,28)(18,32)(20,33)(24,35)(25,34)(27,36)(31,37);(9,21)(14,28)(19,23)(22,34)(26,36)(29,37)(30,32)(33,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26360);