Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #541 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#117 | C2 |
| prime radical | C23×C5 | C23 |
| hypercenter | #117 | D5 |
| hypocenter | C5 | #541 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #117 | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,4,5,3)(7,11)(10,13)(14,17,18,25)(15,20,21,27)(16,23,24,29)(19,26,22,28);(6,8,9,12)(7,10,11,13)(17,25)(20,27)(23,29)(26,28);(14,16,18,24)(15,19,21,22)(17,23,25,29)(20,26,27,28);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)(14,15,18,21)(16,22,24,19)(17,20,25,27)(23,28,29,26)Polycyclic presentation:
2^6,5|a^2=f,b^2=f,c^2=e,d^2=f,b^a=bf,d^a=de,g^a=g^4,d^c=df62 conjugacy classes — view the full character table.
| maximal subgroup of | #10610298, #10610301, #10610308, … |
| maximal normal subgroup of | #10610298, #10610301, #10610308, … |
| maximal quotient of | #10605117, #10605119, #10605121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1190);