Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #24338 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C22×#509 | C22 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20438 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2291 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)(5,6,9,14,10,16,28,30,11,15,22,31,29,7,17,12)(8,19,23,34,26,18,21,36,25,13,27,32,24,33,35,20);(1,2)(3,4)(5,7,10,15,11,16,29,6)(8,20,24,34,25,36,26,32)(9,14,28,12,22,31,17,30)(13,27,33,21,19,23,18,35);(6,12)(7,14)(8,21)(9,17)(10,29)(13,18)(15,30)(16,31)(19,33)(22,28)(23,26)(24,27)(25,35)(32,34);(2,4)(5,8,22,23,10,26,17,21,11,25,9,27,29,24,28,35)(6,13,31,32,16,33,12,20,15,19,14,34,7,18,30,36)Polycyclic presentation:
2^10|a^2=e,b^2=g,d^2=ehj,e^2=i,g^2=j,i^2=j,b^a=beij,c^a=ceh,c^b=cehj,d^a=df,d^b=defgij,d^c=dehij,e^b=ei,e^c=eij,e^d=ej,f^a=fj,f^b=fj,g^a=gj,g^c=gj,g^d=gj,h^d=hj,i^b=ij,i^c=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72250777, #72252658, #72252706, … |
| maximal normal subgroup of | #72250777, #72252658, #72252706, … |
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.5);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.5*F.8*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, 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, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);