Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23×C5 |
| prime radical | C5×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ G | |
82 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)(3,11)(4,15)(5,20)(6,8)(7,12)(9,18)(10,17)(13,31)(14,28)(16,26)(19,27)(21,25)(22,30)(23,32)(24,29)(33,34,35,36,37);(1,3,6,14)(2,7,8,22)(4,16,19,32)(5,21,17,13)(9,25,24,31)(10,26,20,23)(11,27,28,15)(12,29,30,18);(2,8)(7,22)(9,24)(10,20)(11,28)(15,27)(23,26)(25,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1011);