Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C23×C42 |
| commutator | C2×C4 | C23×C4×C8 |
| Frattini, cosocle | C4×C16 | C25 |
| socle | C22 | C23×#346 |
| prime radical | #3444 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 80 points:
(3,9)(7,12)(10,15)(14,16)(17,18,21,30,23,32,45,59,25,34,43,64,22,49,46,63,26,28,48,60,50,61,44,29,24,33,47,62,51,31,20,42)(19,27,37,54,39,56,72,75,41,58,70,80,38,74,65,79,35,52,66,76,68,77,71,53,40,57,73,78,67,55,36,69);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19)(18,27)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,35)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(42,69)(43,70)(44,71)(45,72)(46,65)(47,73)(48,66)(49,74)(50,68)(51,67)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80);(2,6)(7,12)(8,13)(14,16)(18,28)(19,35)(29,59)(30,60)(31,49)(32,61)(33,34)(36,65)(37,66)(38,67)(39,68)(40,41)(42,63)(62,64)(70,73)(71,72);(19,35)(27,52)(36,65)(37,66)(38,67)(39,68)(40,41)(53,75)(54,76)(55,74)(56,77)(57,58)(69,79)(70,73)(71,72)(78,80);(1,3,4,10,5,9,11,15)(2,7,8,14,6,12,13,16)(17,20,22,43,24,44,23,21,26,46,51,47,25,45,50,48)(18,29,31,60,33,63,61,64,28,59,49,30,34,42,32,62)(19,36,38,70,40,71,39,37,35,65,67,73,41,72,68,66)(27,53,55,76,57,79,77,80,52,75,74,54,58,69,56,78)Polycyclic presentation:
2^11|a^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,j^2=k,c^a=ck,c^b=cijk,d^b=dk,e^a=eik,f^e=fk,g^a=gk416 conjugacy classes — view the full character table.
| maximal subgroup of | #72231972 |
| maximal normal subgroup of | #72231972 |
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.6);;
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, 3, 1, F.3*F.11);;
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, 1, F.5*F.9*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);