Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#508 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | D8 |
| commutator, Frattini, cosocle | C8 | C23×C5 |
| socle | C2×C5 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2)(3,12)(4,9)(5,17)(6,21)(7,11)(8,14)(10,19)(13,28)(15,29)(16,27)(18,23)(20,31)(22,26)(24,32)(25,30)(33,34,35,36,37);(1,3)(2,8)(4,13)(5,18)(6,22)(7,15)(9,23)(10,26)(11,25)(12,17)(14,19)(16,30)(20,32)(21,24)(27,28)(29,31);(1,4,7,6)(2,9,11,21)(3,13,15,22)(5,16,20,19)(8,23,25,24)(10,17,27,31)(12,28,29,26)(14,18,30,32)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1009);