Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13048 |
| commutator | #532 | C22×C4 |
| Frattini, cosocle | C2×#532 | C23 |
| socle | C2 | #148170 |
| prime radical | C2×#3481 | C22 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2040 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 2040 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,7,12,6,11)(3,13,15,9,17,27,16,28)(4,8,19,23,20,25,21,24)(14,22,29,26,30,32,18,31);(2,8)(9,26)(10,23)(11,24)(12,25)(13,22)(27,32)(28,31);(1,3)(2,9,12,28)(4,18,20,29)(5,16)(6,15)(7,17)(8,22)(10,13,11,27)(14,19,30,21)(23,31)(24,26)(25,32)Polycyclic presentation:
2^10|a^2=e,c^2=ghi,e^2=j,f^2=i,g^2=j,h^2=j,b^a=bd,c^a=cg,c^b=cf,d^c=dh,e^c=ej,f^a=fhij,f^b=fi,f^d=fi,h^b=hi,h^d=hj,i^a=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72339660, #72339699 |
| maximal normal subgroup of | #72339660, #72339699 |
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, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);