Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33560 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C5 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
1132 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)(9,13)(10,14)(11,15)(12,18)(17,30)(19,31)(20,34)(21,27)(22,28)(23,29)(24,32)(25,33)(26,35)(36,37);(6,8,11,19)(7,12,15,25)(9,20,21,17)(10,18,23,33)(13,26,27,24)(14,16,29,31)(22,35,36,32)(28,34,37,30);(9,21)(10,23)(13,27)(17,20)(18,33)(24,26)(28,37)(30,34);(6,9,11,21)(7,13,15,27)(8,17,19,20)(10,22,23,36)(12,24,25,26)(14,28,29,37)(16,30,31,34)(18,32,33,35)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26548);