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 | #24196 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#13340 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20436 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,10,17,23,29,13,18,8,20,11,7,19,14)(9,21,22,35,25,15,30,33,26,31,24,28,27,32,36,34);(2,3)(5,7,10,18,13,17,11,6)(8,20,19,29,12,16,23,14)(9,21,25,32,26,31,27,15)(22,33,30,35,24,34,36,28);(1,2)(3,4)(5,8,11,23,13,12,10,19)(6,14,17,16,18,29,7,20)(9,24,27,30,26,22,25,36)(15,28,31,34,32,35,21,33);(1,3,4,2)(5,9,11,27,13,26,10,25)(6,15,17,31,18,32,7,21)(8,22,23,36,12,24,19,30)(14,28,16,34,29,35,20,33)Polycyclic presentation:
2^10|a^2=h,b^2=e,c^2=ej,d^2=efgij,e^2=j,f^2=j,h^2=i,i^2=j,b^a=bhij,c^a=cf,c^b=cg,d^a=dij,d^b=dfgi,d^c=defgj,e^a=ej,f^b=fj,f^d=fj,g^d=gj,h^b=hi,h^c=hj,h^d=hj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72321155, #72322277, #72322868, … |
| maximal normal subgroup of | #72321155, #72322277, #72322868, … |
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.5*F.10);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, 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, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);