Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
7026 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 7026 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,24,27)(2,11,40,9,32,38)(3,26,46)(4,51,53)(5,25,7)(8,30,58,29,39,57)(10,35,60,18,47,59)(12,33)(13,41,62,34,56,61)(14,49,64)(15,45,17)(19,43,63)(20,52,22)(31,37)(36,48)(42,55);(1,3,7,17)(2,8,9,29)(4,19,22,14)(5,16)(6,26)(10,34,18,13)(11,31)(12,39)(15,28)(20,50)(21,43)(23,24)(25,45,27,46)(30,33)(32,37)(35,55)(36,41)(38,57,40,58)(42,47)(44,51)(48,56)(49,54)(52,64,53,63)(59,62,60,61);(1,2)(3,13)(4,18)(5,11)(6,12)(7,9)(8,14)(10,22)(15,41)(16,42)(17,34)(19,29)(20,47)(21,48)(23,55)(24,32)(25,38)(26,56)(27,40)(28,33)(30,43)(31,44)(35,51)(36,54)(37,50)(39,49)(45,61)(46,62)(52,60)(53,59)(57,63)(58,64)Polycyclic presentation:
2^4,3,2^5|b^2=j,d^2=j,h^2=j,i^2=j,b^a=bd,c^a=cj,d^a=dj,d^b=dj,e^b=e^2,f^b=fg,f^e=g,g^e=fg,h^b=ij,h^e=i,h^f=hj,h^g=hj,i^b=hj,i^e=hij,i^g=ij,i^h=ij59 conjugacy classes — view the full character table.
| maximal subgroup of | #72227841 |
| maximal normal subgroup of | #72227841 |
| maximal quotient of | #72268154, #72271310, #72271662, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);