Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #251827 |
| commutator, Frattini, cosocle | C22×#127 | C24 |
| LCS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
5014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,9,11,13)(4,12)(6,14,10,16)(7,15)(17,18)(19,26,28,31)(20,27,29,30)(21,24)(22,25)(23,32);(1,2)(3,10)(4,7)(5,8)(6,11)(9,16)(12,15)(13,14)(17,18)(19,20)(21,23)(22,25)(24,32)(26,27)(28,29)(30,31);(1,3)(2,6)(4,13,12,9)(5,11)(7,16,15,14)(8,10)(17,19)(18,20)(21,30,32,27)(22,28)(23,31,24,26)(25,29);(7,15)(14,16)(17,20)(18,19)(21,31)(22,29)(23,27)(24,30)(25,28)(26,32)Polycyclic presentation:
2^11|a^2=h,c^2=f,g^2=k,c^a=ce,c^b=cjk,d^a=dg,d^b=df,e^a=ej,e^c=eh,e^d=ei,f^a=fh,f^e=fj,g^b=ghk,g^c=gik,g^d=gk,g^e=gk,g^f=gk,h^c=hj,h^d=hk,i^a=ik,i^b=ij,i^c=ik62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338775, #72338830, #72339103, … |
| maximal normal subgroup of | #72338775, #72338830, #72339103, … |
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.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 7, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*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, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);