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×C22⋊Q8 | C2 |
| prime radical | C23×C5 | C23 |
| hypercenter | C22⋊Q8 | D5 |
| hypocenter | C5 | #541 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(8,12)(10,13)(15,21)(19,27)(20,29)(26,28);(6,8,9,12)(7,10,11,13)(14,17,18,25)(15,20,21,29)(16,22,23,24)(19,26,27,28);(14,16,18,23)(15,19,21,27)(17,24,25,22)(20,28,29,26);(2,3)(4,5)(6,7,9,11)(8,10,12,13)(14,15)(16,19)(17,20)(18,21)(22,26)(23,27)(24,28)(25,29)Polycyclic presentation:
2^6,5|a^2=f,b^2=e,c^2=ef,d^a=de,g^a=g^4,c^b=ce,d^c=df50 conjugacy classes — view the full character table.
| maximal subgroup of | #10611868, #10611873, #10611908, … |
| maximal normal subgroup of | #10611868, #10611873, #10611908, … |
| maximal quotient of | #10605599, #10605602, #10605623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1297);