Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #30396 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#33541 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3152 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)(6,12,15,16)(8,20)(13,25,29,27)(14,30,32,26)(18,22)(21,23)(28,36,31,35)(33,34);(5,6,8,13,10,16,21,25,11,15,20,29,9,12,23,27)(7,14,18,31,19,30,34,36,17,32,22,28,24,26,33,35);(2,4)(6,13,15,29)(7,17)(8,21)(9,10)(12,25,16,27)(14,31,32,28)(18,33)(20,23)(22,34)(26,36,30,35);(1,2,3,4)(5,7,10,19,11,17,9,24)(6,14,16,30,15,32,12,26)(8,22,21,33,20,18,23,34)(13,28,25,35,29,31,27,36)Polycyclic presentation:
2^10|a^2=h,b^2=e,c^2=h,d^2=fhi,e^2=i,g^2=j,i^2=j,b^a=bg,c^b=ceg,d^a=df,d^b=dhj,d^c=dfghi,e^a=ej,e^c=ei,e^d=ej,f^b=fj,g^c=gj,g^d=gj,h^b=hj,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72250127, #72250414, #72257268, … |
| maximal normal subgroup of | #72250127, #72250414, #72257268, … |
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.5);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6*F.8*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, 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.5*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, 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, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);