Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #24033 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | #6443272 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,9,14,26,7,11,19,12,17,25,32,10,15,27,16)(8,13,21,28,22,18,23,33,20,31,34,36,24,30,35,29);(5,7,10,17,12,15,26,6)(8,18,24,31,20,30,22,13)(9,14,27,19,25,32,11,16)(21,28,35,33,34,36,23,29);(1,2)(3,4)(6,7,17,15)(8,20)(9,25)(11,27)(13,18,31,30)(14,16,32,19)(22,24)(28,29,36,33);(1,3,4,2)(5,8)(6,13)(7,18)(9,21)(10,24)(11,23)(12,20)(14,28)(15,30)(16,29)(17,31)(19,33)(22,26)(25,34)(27,35)(32,36)Polycyclic presentation:
2^10|a^2=e,b^2=ehij,c^2=ehj,d^2=fgj,e^2=ij,f^2=j,g^2=j,h^2=i,i^2=j,b^a=bh,c^a=cg,c^b=cf,d^a=dfgj,d^c=defghj,e^b=ei,e^c=ej,f^b=fj,f^d=fj,g^b=gj,g^d=gj,h^a=hj,h^b=hi,h^c=hj,i^b=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72321390, #72322146, #72322519, … |
| maximal normal subgroup of | #72321390, #72322146, #72322519, … |
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.5);;
gap> SetPower(coll, 2, F.5*F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.5*F.8*F.10);;
gap> SetPower(coll, 4, F.6*F.7*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*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, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);