Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9379 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
492 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)(6,7,10,17,23,33,8,20)(9,25,28,35,37,16,24,32)(11,18,21,13,12,19,22,15)(14,31,29,27,34,36,30,26);(7,13,18,17)(8,21)(9,26)(10,22)(14,29,25,35)(15,19,20,33)(16,32,34,30)(24,28)(27,37)(31,36);(7,14)(8,22)(9,27)(10,21)(13,29)(15,30)(16,19)(17,35)(18,25)(20,32)(24,28)(26,37)(31,36)(33,34)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2367);