Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#2177 |
| commutator | C16 | C23×C8 |
| Frattini, cosocle | C4×C16 | C24 |
| prime radical | C2×#2262 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ #11399 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,6,5)(3,7,4,8)(11,21)(12,28)(13,31)(14,33)(17,38)(18,42)(19,43)(20,37)(22,45)(23,46)(24,30)(25,48)(26,59)(27,54)(29,40)(32,57)(34,64)(35,61)(36,62)(39,51)(41,70)(44,68)(47,49)(50,63)(52,69)(53,60)(55,56)(58,72)(65,71)(66,67);(1,3,5,8,6,4,2,7)(9,10)(11,22,25,39)(12,29)(13,19)(14,20)(15,16)(17,23,35,27)(18,30)(21,45,48,51)(24,42)(26,52,58,66)(28,40)(31,43)(32,63)(33,37)(34,36)(38,46,61,54)(41,55,65,47)(44,60)(49,70,56,71)(50,57)(53,68)(59,69,72,67)(62,64);(1,4,5,7,6,3,2,8)(10,16)(11,23,25,27)(12,30)(17,39,35,22)(19,36)(20,37)(21,46,48,54)(24,28)(26,55,58,47)(32,60)(38,51,61,45)(41,66,65,52)(43,62)(49,59,56,72)(53,57)(67,71,69,70);(9,11,24,56,13,26,32,46,33,23,53,59,64,55,12,21,15,25,28,49,34,58,60,54,14,27,57,72,31,47,30,48)(10,17,40,69,19,41,44,45,37,22,50,70,62,52,18,38,16,35,42,67,36,65,63,51,20,39,68,71,43,66,29,61)Polycyclic presentation:
2^10|a^2=h,b^2=f,c^2=f,d^2=egj,e^2=gi,f^2=h,g^2=ij,i^2=j,c^b=cj,d^a=de,d^b=dij,d^c=dij,e^a=eg,e^b=ej,e^c=ej,f^d=fj,g^a=gi,i^a=ij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72232013, #72321506, #72321704, … |
| maximal normal subgroup of | #72232013, #72321506, #72321704, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.5*F.7*F.10);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);