Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #464 |
112 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4,5,3)(6,7)(8,9)(11,16)(12,18)(13,21)(15,22)(19,20)(23,25);(6,7,8,9)(10,12,14,20)(11,15,16,23)(13,18,21,19)(17,25,24,22);(2,3)(4,5)(6,7,8,9)(10,11)(12,15)(13,17)(14,16)(18,25)(19,22)(20,23)(21,24)Polycyclic presentation:
2^6,5|a^2=e,b^2=ef,d^2=f,c^a=cef,g^a=g^4,c^b=cd,d^b=df,d^c=df47 conjugacy classes — view the full character table.
| maximal subgroup of | #10610490, #10610494, #10610498, … |
| maximal normal subgroup of | #10610490, #10610494, #10610498, … |
| maximal quotient of | #10599203, #10599204, #10599207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 394);