Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72235729 |
| commutator | #2308 | C22×C4 |
| Frattini, cosocle | C2×#2308 | C23 |
| prime radical | #72237746 | C2 |
| LCS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C26 ◁ C2×#2308 ◁ G | |
| LECS | G ▷ C2×#2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
4082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,10,19)(3,11,14,16)(5,18)(8,23,26,12)(9,28,24,22)(15,32,20,31)(17,29)(21,30,27,25);(1,2,5,19)(3,12,20,25,14,23,15,30)(4,9,29,28)(6,10,18,7)(8,11,21,31,26,16,27,32)(13,24,17,22);(1,3,13,11,6,14,4,16)(2,8,24,25,10,26,9,30)(5,20,17,31,18,15,29,32)(7,21,22,12,19,27,28,23)Polycyclic presentation:
2^11|a^2=e,b^2=efgik,c^2=dehjk,d^2=k,f^2=ik,b^a=bf,c^a=cd,c^b=cdfghij,d^a=dj,d^b=dh,d^c=dk,e^b=ei,e^c=ejk,f^a=fk,f^b=fk,f^c=fhij,f^d=fj,g^a=gik,g^b=gik,g^c=gik,g^d=gk,h^a=hj,h^b=hjk,h^c=hj,h^f=hk,i^c=ik,j^b=jk74 conjugacy classes — view the full character table.
| maximal subgroup of | #72237948, #72237974, #72238600, … |
| maximal normal subgroup of | #72237948, #72237974, #72238600, … |
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, 1, F.5);;
gap> SetPower(coll, 2, F.5*F.6*F.7*F.9*F.11);;
gap> SetPower(coll, 3, F.4*F.5*F.8*F.10*F.11);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);