Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #120014 |
| commutator | C23×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C24×D4 | C23 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1 | |
2283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,9)(5,11,6,20)(7,10)(8,15,22,16)(13,25)(14,26)(17,24)(18,23,19,30)(21,27,31,29)(28,32);(1,3,7,14)(2,8)(4,13)(5,15,6,16)(9,21)(10,22)(11,12)(17,28)(18,29)(19,27)(20,26)(23,32)(24,31)(25,30);(1,4)(2,9,10,24)(3,13,14,28)(5,19)(6,18)(7,17)(8,21)(11,23,20,30)(12,25)(15,29,16,27)(22,31)(26,32)Polycyclic presentation:
2^10|a^2=d,b^2=fgj,c^2=eh,f^2=j,g^2=j,b^a=bf,c^a=cdhj,c^b=cde,d^b=di,e^a=ej,e^b=ej,e^c=ej,f^a=fij,f^b=fh,f^c=fij,f^d=fj,g^a=ghi,g^b=gh,g^d=gj,h^c=hj,i^a=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72256106, #72256121, #72256135, … |
| maximal normal subgroup of | #72256106, #72256121, #72256135, … |
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.4);;
gap> SetPower(coll, 2, F.6*F.7*F.10);;
gap> SetPower(coll, 3, F.5*F.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);