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×#2263 | 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 | |
656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,6,5)(3,7,4,8)(11,21)(12,27)(13,30)(14,32)(17,38)(18,42)(19,44)(20,37)(22,46)(23,52)(24,29)(25,57)(26,50)(28,55)(31,53)(33,59)(34,43)(35,62)(36,63)(39,65)(40,64)(41,70)(45,60)(47,48)(49,68)(51,58)(54,56)(61,69)(66,67)(71,72);(9,10)(11,17)(12,18)(13,19)(14,20)(15,16)(21,38)(22,49)(23,40)(24,34)(25,41)(26,39)(27,42)(28,35)(29,43)(30,44)(31,45)(32,37)(33,36)(46,68)(47,66)(48,67)(50,65)(51,72)(52,64)(53,60)(54,69)(55,62)(56,61)(57,70)(58,71)(59,63);(1,3,5,8,6,4,2,7)(10,16)(11,22,24,26)(12,28)(17,39,34,49)(19,36)(20,37)(21,46,29,50)(23,53)(25,51,56,47)(27,55)(31,52)(38,65,43,68)(41,66,61,72)(44,63)(48,57,58,54)(67,69,71,70);(1,4,5,7,6,3,2,8)(9,11,23,54,33,56,28,21,32,22,27,58,30,47,31,50,15,24,53,57,13,25,12,29,14,26,55,48,59,51,52,46)(10,17,40,69,36,61,35,38,37,49,42,71,44,66,45,65,16,34,60,70,19,41,18,43,20,39,62,67,63,72,64,68)Polycyclic presentation:
2^10|a^2=h,c^2=f,d^2=efgi,e^2=gi,f^2=h,g^2=ij,i^2=j,c^b=cj,d^a=de,d^c=dij,e^a=eg,e^c=ej,e^d=ej,f^d=fj,g^a=gi,i^a=ij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72232013, #72322276, #72322300, … |
| maximal normal subgroup of | #72232013, #72322276, #72322300, … |
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, 3, F.6);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.9);;
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, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, 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);