Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23412 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1948 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)(9,23)(10,15)(11,16)(12,17)(13,21)(14,24)(19,27)(20,28)(22,32)(25,31)(26,30)(29,35)(33,34)(36,37);(6,8)(7,13)(9,19)(10,20)(11,21)(12,22)(14,29)(15,18)(16,32)(17,28)(23,33)(24,34)(25,36)(26,35)(27,31)(30,37);(6,9)(7,14)(8,19)(10,24)(11,25)(12,26)(13,27)(15,30)(16,23)(17,31)(18,33)(20,34)(21,36)(22,35)(28,29)(32,37);(6,10,12,11)(7,15,17,16)(8,20,22,21)(9,24,26,25)(13,18,28,32)(14,30,31,23)(19,34,35,36)(27,33,29,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16400);