Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#2177 |
| commutator | C2×C16 | C24×C4 |
| Frattini, cosocle | C4×C16 | C25 |
| socle | C22 | C22×#2177 |
| prime radical | #72234362 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C82 ◁ C22×#1311 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2,5,6)(3,7,10,12)(4,8,11,13)(9,14,15,16)(17,18)(19,29)(20,36)(21,41)(22,44)(23,24)(25,32)(26,39)(27,38)(28,42)(30,46)(31,54)(33,53)(34,67)(35,68)(37,48)(40,73)(43,74)(45,64)(47,71)(49,62)(50,78)(51,60)(52,58)(55,65)(56,63)(57,77)(59,69)(61,76)(66,80)(70,72)(75,79);(2,6)(7,12)(8,13)(14,16);(18,24)(25,45)(26,46)(27,47)(28,44)(29,53)(36,48)(41,74)(49,51)(50,75)(52,69)(54,68)(55,77)(56,76)(67,80)(70,73);(17,19,30,57,21,34,40,62,42,31,59,78,71,61,20,32,23,33,39,65,43,66,72,60,22,35,58,79,38,63,37,64)(18,25,48,76,27,50,52,54,44,49,70,67,74,77,26,29,24,45,36,56,47,75,69,68,28,51,73,80,41,55,46,53);(1,3,4,9,5,10,11,15)(2,7,8,14,6,12,13,16)(19,31,33,35)(20,37)(25,49,45,51)(26,46)(29,54,53,68)(30,39)(32,62,64,60)(34,61,66,63)(36,48)(40,72)(50,77,75,55)(52,69)(56,67,76,80)(57,79,65,78)(58,59)(70,73)Polycyclic presentation:
2^11|a^2=i,d^2=fhk,e^2=g,f^2=hj,g^2=i,h^2=jk,j^2=k,b^a=bi,c^a=ck,d^a=df,e^d=ej,f^a=fh,f^e=fk,g^d=gk,h^a=hj,j^a=jk224 conjugacy classes — view the full character table.
| maximal subgroup of | #72321290, #72322589, #72323032, … |
| maximal normal subgroup of | #72321290, #72322589, #72323032, … |
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.9);;
gap> SetPower(coll, 4, F.6*F.8*F.11);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);