Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C5 ⋊ #452 |
152 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 47 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)(7,9)(10,13,15,14)(11,17,18,22)(12,20,21,23)(16,24,25,19);(10,12)(11,16)(13,20)(14,23)(15,21)(17,24)(18,25)(19,22);(2,3)(4,5)(6,7)(8,9)(10,11,15,18)(12,19,21,24)(13,17,14,22)(16,23,25,20)Polycyclic presentation:
2^6,5|a^2=f,c^2=f,e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^4,e^b=ef59 conjugacy classes — view the full character table.
| maximal subgroup of | #10610576, #10610578, #10610580, … |
| maximal normal subgroup of | #10610576, #10610578, #10610580, … |
| maximal quotient of | #10602369, #10602378, #10602399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 866);