Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#1333 |
| commutator | C2×C16 | C23×C8 |
| Frattini, cosocle | #1311 | C24 |
| socle | C22 | C22×#1333 |
| prime radical | #7623228 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C82 ◁ C22×#1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 80 points:
(2,6,8,11,7,14,9,15)(3,10)(12,16)(17,18,20,28,22,30,44,29,24,32,45,64,23,54,21,48,26,34,47,66,53,69,49,68,25,33,46,65,52,31,50,63)(19,35,37,60,39,71,74,80,41,72,75,56,40,78,43,79,36,70,73,61,55,67,77,59,42,62,76,27,51,57,38,58);(1,2,4,8,5,7,13,9)(3,11,12,14,10,15,16,6)(17,19,26,36)(18,27)(20,43,47,38)(21,37,50,73)(22,51,53,40)(23,55,52,39)(24,42,25,41)(28,62)(29,67)(30,59)(31,58)(32,61)(33,60)(34,56)(35,65)(44,75,49,76)(45,74,46,77)(48,78)(54,79)(57,63)(64,70)(66,72)(68,71)(69,80);(2,7)(6,14)(8,9)(11,15)(19,36)(27,56)(35,70)(37,73)(38,43)(39,55)(40,51)(41,42)(57,78)(58,79)(59,80)(60,61)(62,72)(67,71)(74,77)(75,76);(19,36)(27,56)(35,70)(37,73)(38,43)(39,55)(40,51)(41,42)(57,78)(58,79)(59,80)(60,61)(62,72)(67,71)(74,77)(75,76)Polycyclic presentation:
2^11|a^2=e,b^2=gh,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,j^2=k,b^a=bf,c^b=cijk,d^b=dk,e^b=eh,f^a=fjk,f^b=fhj,f^e=fk,g^b=gj,h^a=hk,h^b=hjk,i^b=ik,j^b=jk224 conjugacy classes — view the full character table.
| maximal subgroup of | #72322541, #72322903, #72323108 |
| maximal normal subgroup of | #72322541, #72322903, #72323108 |
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.7*F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);