Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D16 |
| commutator, Frattini, cosocle | C2×C16 | C25 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ C4×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,7)(3,8,4,6)(11,21)(12,27)(13,28)(14,31)(17,36)(18,40)(19,41)(20,35)(22,43)(23,30)(24,44)(25,54)(26,55)(29,51)(32,58)(33,37)(34,59)(38,66)(39,67)(42,65)(45,46)(47,53)(48,71)(49,63)(50,70)(52,72)(56,57)(60,68)(61,62)(64,69);(1,3,5,4)(2,6,7,8)(9,10)(11,22)(12,18)(13,19)(14,20)(15,16)(17,24)(21,43)(23,37)(25,48)(26,49)(27,40)(28,41)(29,42)(30,33)(31,35)(32,34)(36,44)(38,52)(39,53)(45,69)(46,64)(47,67)(50,62)(51,65)(54,71)(55,63)(56,60)(57,68)(58,59)(61,70)(66,72);(10,16)(17,22)(18,33)(19,34)(20,35)(36,43)(37,40)(38,48)(39,49)(41,59)(42,60)(61,69)(62,64)(63,67)(65,68)(66,71);(9,11,23,50,13,25,29,47,31,53,57,54,58,70,12,21,15,24,27,46,32,52,56,55,14,26,51,72,28,45,30,44)(10,17,37,64,19,38,42,63,35,49,68,66,59,69,18,36,16,22,40,62,34,48,60,67,20,39,65,71,41,61,33,43);(1,4,5,3)(2,6,7,8)(11,24)(17,22)(21,44)(25,52)(26,53)(36,43)(38,48)(39,49)(45,70)(46,50)(47,55)(54,72)(61,69)(62,64)(63,67)(66,71)Polycyclic presentation:
2^10|a^2=g,b^2=g,d^2=fhj,e^2=g,f^2=hi,h^2=ij,i^2=j,b^a=bg,c^b=cj,d^a=df,d^b=dj,e^d=ej,f^a=fh,h^a=hi,i^a=ij160 conjugacy classes — view the full character table.
| maximal subgroup of | #72232026, #72321842, #72321873, … |
| maximal normal subgroup of | #72232026, #72321842, #72321873, … |
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.7);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.6*F.8*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);