Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #31680 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#24281 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#2929 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,12)(7,17)(8,22)(9,11)(13,29)(14,33)(18,24)(19,32)(20,35)(21,34)(23,28)(25,26)(27,30)(31,36);(6,12)(7,18)(13,27)(14,28)(15,16)(17,24)(19,36)(20,21)(23,33)(29,30)(31,32)(34,35);(5,6,9,16,10,12,11,15)(7,19,20,29,18,36,21,30)(8,23,25,28,26,33,22,14)(13,17,32,34,27,24,31,35);(2,4)(5,7,8,24,9,20,25,35,10,18,26,17,11,21,22,34)(6,13,28,30,15,31,23,36,12,27,14,29,16,32,33,19)Polycyclic presentation:
2^10|c^2=ghi,d^2=efgi,e^2=ij,f^2=j,i^2=j,c^a=cg,c^b=ch,d^a=de,d^b=dgh,d^c=defj,e^a=ei,e^c=ei,e^d=ej,f^a=fj,f^c=fj,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 | #72257287, #72257324, #72257453, … |
| maximal normal subgroup of | #72257287, #72257324, #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, 3, F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 1, 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, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, 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, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);