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×#33360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#99 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×D4 | C24×C5 |
| prime radical | C5×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×Q8 ◁ C2×C5×2−1+4 ◁ G | |
1188 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)(8,13)(9,14)(10,15,23,12)(11,16)(17,26)(18,27,33,24)(19,28)(20,29)(21,30,35,25)(22,31)(32,37,34,36);(7,12)(10,23)(13,24)(14,25)(15,16)(18,33)(21,35)(26,36)(27,28)(29,37)(30,31)(32,34);(6,8,11,19)(7,13,16,28)(9,20,22,17)(10,18,23,33)(12,24,15,27)(14,29,31,26)(21,34,35,32)(25,37,30,36);(6,9,11,22)(7,14,16,31)(8,17,19,20)(10,21,23,35)(12,25,15,30)(13,26,28,29)(18,32,33,34)(24,36,27,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26348);