Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #23764 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C22×#472 | C22 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,7)(8,19,23,33)(9,25)(10,15)(12,27,16,28)(13,26)(17,32)(20,30,36,29)(21,31)(22,35,24,34);(1,2,3,4)(5,6,10,14,11,7,15,18)(8,12,22,29,23,16,24,30)(9,17,25,26,21,13,31,32)(19,27,34,36,33,28,35,20);(2,4)(5,7,15,18,11,6,10,14)(8,20,22,28,23,36,24,27)(9,26,31,13,21,32,25,17)(12,19,30,34,16,33,29,35);(5,8,21,33,10,24,31,34,11,23,9,19,15,22,25,35)(6,12,13,28,14,30,32,36,7,16,17,27,18,29,26,20)Polycyclic presentation:
2^10|a^2=h,b^2=f,c^2=ghij,d^2=eghj,e^2=i,f^2=j,i^2=j,b^a=bg,c^a=cg,c^b=cfghi,d^a=de,d^b=dghj,d^c=deghij,e^a=ei,e^b=ej,e^c=ei,f^a=fj,f^c=fj,f^d=fj,g^b=gj,g^c=gj,g^d=gj,h^d=hj,i^a=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72250361, #72257230, #72257453, … |
| maximal normal subgroup of | #72250361, #72257230, #72257453, … |
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.6);;
gap> SetPower(coll, 3, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.5*F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*F.8*F.9*F.10);;
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);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, 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, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, 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);