Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72235716 |
| commutator, Frattini, cosocle | #2308 | C24 |
| prime radical | #72216464 | C2 |
| LCS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #35346 ◁ G | |
| LECS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
5907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,11)(9,25)(12,26)(18,23)(19,30)(22,32);(1,2)(3,8)(4,9)(5,7)(6,10)(11,24)(12,22)(13,18)(14,19)(15,25)(16,30)(17,20)(21,28)(23,27)(26,32)(29,31);(2,7)(3,12)(8,21)(9,19)(10,20)(11,26)(13,29)(18,22)(23,32)(24,31)(25,30)(27,28);(1,3)(2,8,10,24)(4,12)(5,13)(6,11)(7,18,20,23)(9,22,25,32)(14,28)(15,26)(16,29)(17,27)(19,21,30,31)Polycyclic presentation:
2^11|d^2=j,f^2=i,b^a=bg,c^b=cf,d^a=dhjk,d^b=dik,d^c=de,e^a=ej,e^b=eh,f^a=fi,f^b=fi,f^c=fi,f^d=fhjk,f^e=fjk,g^c=gi,g^d=gk,g^e=gk,h^c=hj,h^f=hk,i^d=ik,j^b=jk68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237772, #72337716, #72337718, … |
| maximal normal subgroup of | #72237772, #72337716, #72337718, … |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);