Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#497 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C5 |
| prime radical | C2×C5×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 25 points:
(2,4)(5,6,9,12)(7,13,14,20)(8,16,17,11)(10,18,19,15)(21,22,23,24,25);(5,7,8,15,9,14,17,18)(6,10,11,13,12,19,16,20);(1,2,3,4)(7,14)(10,19)(13,20)(15,18)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 998);