Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #31681 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | #411837 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20385 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(7,17)(19,23)(21,26)(35,36);(1,2,3,4)(5,6,10,16,11,12,9,15)(7,18,21,27,17,32,26,13)(8,22,20,31,25,14,24,29)(19,34,35,30,23,33,36,28);(2,4)(6,12)(7,19,17,23)(8,20)(9,10)(13,28,27,30)(14,31)(18,33,32,34)(21,36,26,35)(22,29)(24,25);(5,7,20,36,9,26,8,23,11,17,24,35,10,21,25,19)(6,13,29,34,15,32,14,28,12,27,31,33,16,18,22,30)Polycyclic presentation:
2^10|b^2=fhj,c^2=hi,d^2=ehj,e^2=i,h^2=j,i^2=j,b^a=bf,c^b=cfg,d^a=dghij,d^b=dgh,d^c=de,e^b=ej,e^c=ei,e^d=ej,f^d=fj,g^b=gj,g^d=gj,h^c=hj,h^d=hj,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72257434, #72257623, #72257690, … |
| maximal normal subgroup of | #72257434, #72257623, #72257690, … |
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, 2, F.6*F.8*F.10);;
gap> SetPower(coll, 3, F.8*F.9);;
gap> SetPower(coll, 4, F.5*F.8*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);