Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×D16 ⋊ C2 |
| Sylow | C5 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5×D8 |
| commutator, cosocle | C5×C8 | C23 |
| Frattini | C8 | C22×D5 |
| Fitting | C5×D16 | C2 |
| socle | C2×C5 | C2×D8 |
| hypercenter | D16 | D5 |
| hypocenter | C5 | #508 |
| LCS | G ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D16 | |
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:
(1,2,4,5,3)(7,13)(8,17)(9,21)(10,22)(12,25)(14,31)(16,30)(18,19)(20,36)(23,37)(24,35)(26,33)(27,34)(28,32);(6,8)(7,12)(9,17)(10,24)(11,20)(13,16)(14,30)(15,33)(18,21)(19,22)(23,36)(25,28)(26,32)(27,29)(31,34)(35,37);(2,3)(4,5)(6,7)(8,16)(9,14)(10,15)(11,13)(12,20)(17,25)(18,34)(19,33)(21,32)(22,29)(23,28)(24,27)(26,35)(30,36)(31,37)Polycyclic presentation:
2^6,5|d^2=ef,e^2=f,b^a=bf,c^a=cf,g^a=g^4,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #10598679, #10598680, #10600169, … |
| maximal normal subgroup of | #10598679, #10598680, #10600169, … |
| maximal quotient of | #10599968, #10599973, #10599977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 539);