Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18240 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
1056 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)(7,12)(9,19)(13,24)(14,23)(15,20)(17,28)(21,33)(25,30)(26,35)(27,34)(31,32)(36,37);(6,7)(8,13)(9,20)(10,15)(11,14)(12,19)(16,25)(17,30)(18,26)(21,34)(22,27)(23,33)(24,28)(29,37)(31,35)(32,36);(8,16)(9,21)(12,23)(13,25)(17,31)(18,29)(19,33)(20,34)(24,36)(26,37)(28,32)(30,35);(6,8,11,18)(7,13,14,26)(9,17,21,32)(10,16,22,29)(12,24,23,35)(15,25,27,37)(19,28,33,31)(20,30,34,36)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11228);