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 | #32002 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | #411708 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#2929 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(13,26)(17,29)(32,36)(33,35);(2,3)(6,11)(7,16,19,23)(8,21)(9,12)(13,29,26,17)(14,31)(18,24)(20,34,28,30)(22,27)(32,33,36,35);(1,2)(3,4)(5,6,12,15,10,11,9,25)(7,17,28,33,19,29,20,35)(8,22,18,27,24,31,21,14)(13,30,32,16,26,34,36,23);(1,3,4,2)(5,7,18,30,12,28,24,16,10,19,21,34,9,20,8,23)(6,13,31,33,25,36,27,17,11,26,22,35,15,32,14,29)Polycyclic presentation:
2^10|b^2=h,c^2=ghij,d^2=ehij,e^2=i,i^2=j,c^a=cf,c^b=cfg,d^a=dfg,d^b=de,d^c=deghj,e^b=ei,e^c=ei,e^d=ej,f^d=fj,g^c=gj,g^d=gj,h^d=hj,i^b=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72257533, #72257552, #72257582, … |
| maximal normal subgroup of | #72257533, #72257552, #72257582, … |
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, 2, F.8);;
gap> SetPower(coll, 3, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.5*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, 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, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);