Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
1546 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 1546 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(6,12,10)(7,11,8)(13,22,16)(14,21,19)(17,25,26)(20,28,24);(8,11)(9,12)(13,14,17,20)(15,21,23,28)(16,24,26,19)(18,25,27,22);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)Polycyclic presentation:
2^4,3,2^5|b^2=j,f^2=j,g^2=j,c^a=cd,e^b=e^2,f^a=fi,f^b=fg,f^e=g,g^a=gh,g^b=gj,g^e=fgj,g^f=gj,h^e=hi,i^b=hi,i^e=h57 conjugacy classes — view the full character table.
| maximal subgroup of | #72227987 |
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, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);