Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23884 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1684 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)(9,22)(10,16)(11,14)(12,17)(13,20)(15,23)(19,27)(21,29)(24,31)(25,30)(26,28)(32,33)(34,37)(35,36);(6,8)(7,13)(9,19)(10,26)(11,20)(12,21)(14,29)(15,27)(16,18)(17,28)(22,32)(23,35)(24,33)(25,34)(30,36)(31,37);(7,14)(10,11)(13,18)(15,30)(16,17)(20,26)(22,31)(23,24)(27,32)(28,29)(33,35)(36,37);(6,9,12,25)(7,15,17,31)(8,19,21,34)(10,24,11,23)(13,27,28,37)(14,30,16,22)(18,32,29,36)(20,35,26,33)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16872);