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 | #31680 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C24×D8 | C22 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20385 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3058 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)(7,16)(18,30)(20,25)(21,22);(1,2)(3,4)(5,6,9,14,10,11,26,15)(7,17,21,31,16,35,22,36)(8,23,24,13,27,34,19,29)(12,30,32,25,28,18,33,20);(2,4)(6,11)(7,18)(8,24)(9,26)(12,31)(13,34)(16,30)(17,32)(19,27)(20,22)(21,25)(23,29)(28,36)(33,35);(5,7,19,30,9,21,8,25,10,16,24,18,26,22,27,20)(6,12,23,31,15,33,29,17,11,28,34,36,14,32,13,35)Polycyclic presentation:
2^10|b^2=g,d^2=eghi,e^2=ij,g^2=j,h^2=j,i^2=j,b^a=bf,c^b=cfghj,d^a=dgi,d^b=de,d^c=degj,e^b=ei,e^c=ei,e^d=ej,f^d=fj,g^c=gj,h^b=hj,h^c=hj,h^d=hj,i^b=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72257573, #72257623, #72257738, … |
| maximal normal subgroup of | #72257573, #72257623, #72257738, … |
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.7);;
gap> SetPower(coll, 4, F.5*F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*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, 8, 2, F.8*F.10);;
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> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);