Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24408 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1364 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)(8,16,18,12)(9,13)(10,15)(11,14)(17,26,35,34)(19,32,23,27)(20,28,36,37)(21,30)(22,29)(24,33)(25,31);(6,8)(7,12)(9,20)(10,23)(11,18)(13,28)(14,16)(15,32)(17,22)(19,25)(21,36)(24,35)(26,29)(27,31)(30,37)(33,34);(6,9)(7,13)(8,17)(10,24)(11,21)(12,26)(14,30)(15,33)(16,34)(18,35)(19,36)(20,23)(22,25)(27,37)(28,32)(29,31);(6,8)(7,12)(9,17)(10,19)(11,18)(13,26)(14,16)(15,27)(20,22)(21,35)(23,25)(24,36)(28,29)(30,34)(31,32)(33,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17396);