Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#519 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C24 |
| commutator | C2 | C23×C4×C5 |
| Frattini, cosocle | C22 | C24×C5 |
| prime radical | C23×C5 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,23,24,25);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19);(6,12)(10,13)(11,20)(18,19)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1521);