Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#2177 |
| commutator | C16 | C24×C8 |
| Frattini, cosocle | C4×C16 | C25 |
| prime radical | C2×#33982 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#131 ◁ #479899 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,4)(3,6,8,7)(11,21)(12,27)(13,29)(14,31)(17,37)(18,42)(19,43)(20,36)(22,28)(23,45)(24,47)(25,55)(26,52)(30,50)(32,57)(33,58)(34,38)(35,60)(39,62)(40,70)(41,68)(44,66)(46,48)(49,53)(51,56)(54,71)(59,72)(61,67)(63,64)(65,69);(9,10)(11,17)(12,18)(13,19)(14,20)(15,16)(21,37)(22,38)(23,39)(24,33)(25,40)(26,41)(27,42)(28,34)(29,43)(30,44)(31,36)(32,35)(45,62)(46,63)(47,58)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,59)(55,70)(56,61)(57,60)(71,72);(10,16)(17,33)(18,34)(19,35)(20,36)(37,58)(38,42)(39,41)(40,59)(43,60)(44,61)(62,68)(63,69)(64,65)(66,67)(70,72);(9,11,22,49,13,25,30,45,31,23,51,55,57,53,12,21,15,24,27,48,32,54,56,52,14,26,50,71,29,46,28,47)(10,17,38,65,19,40,44,62,36,39,67,70,60,69,18,37,16,33,42,64,35,59,61,68,20,41,66,72,43,63,34,58);(1,3,4,7,5,8,2,6)(11,23,24,26)(12,28)(17,39,33,41)(18,34)(21,45,47,52)(22,27)(25,53,54,46)(30,56)(37,62,58,68)(38,42)(40,69,59,63)(44,61)(48,55,49,71)(50,51)(64,70,65,72)(66,67)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,c^b=ck,d^a=df,e^d=ej,f^a=fh,f^e=fk,g^d=gk,h^a=hj,j^a=jk260 conjugacy classes — view the full character table.
| maximal subgroup of | #72321544, #72321722, #72322589, … |
| maximal normal subgroup of | #72321544, #72321722, #72322589, … |
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, 3, 2, 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);