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 | #30396 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | #498878 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20438 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12,15,27)(7,17)(8,20)(10,11)(13,25,16,22)(14,29,31,26)(18,33)(21,23)(28,34,30,36)(32,35);(6,13,15,16)(8,21)(12,25,27,22)(14,30,31,28)(18,32)(20,23)(26,34,29,36)(33,35);(2,4)(5,6)(7,14)(8,22)(9,15)(10,16)(11,13)(12,23)(17,31)(18,34)(19,28)(20,27)(21,25)(24,30)(26,33)(29,35)(32,36);(1,2,3,4)(5,7,10,24,9,17,11,19)(6,14,13,28,15,31,16,30)(8,18,23,33,21,32,20,35)(12,26,22,34,27,29,25,36)Polycyclic presentation:
2^10|a^2=h,b^2=h,d^2=fgj,e^2=i,g^2=j,i^2=j,c^a=ce,c^b=cg,d^a=dgi,d^b=dh,d^c=df,e^a=ei,e^b=ej,e^c=eij,f^d=fj,g^a=gj,g^c=gj,g^d=gj,h^c=hj,i^a=ij,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72250127, #72250414, #72257219, … |
| maximal normal subgroup of | #72250127, #72250414, #72257219, … |
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.8);;
gap> SetPower(coll, 4, F.6*F.7*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, 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, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);