Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #122038 |
| 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 | |
2129 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)(4,17,19,28)(5,21)(6,10)(7,11)(8,14)(9,20,27,22)(13,29,18,32)(15,24)(16,25)(23,30,26,31);(1,3,6,16,7,14,5,15)(2,8,10,25,11,12,21,24)(4,18,20,30,19,13,22,31)(9,26,17,32,27,23,28,29);(1,4,6,20)(2,9,10,28)(3,13,15,30)(5,22,7,19)(8,23,24,29)(11,27,21,17)(12,26,25,32)(14,18,16,31)Polycyclic presentation:
2^10|a^2=e,b^2=g,c^2=deg,f^2=j,g^2=j,b^a=bd,c^a=cdfj,c^b=cefgij,d^b=di,d^c=dh,e^c=ej,f^a=fj,f^b=fh,f^c=fhi,f^d=fj,g^a=gi,g^c=ghj,g^d=gj,h^a=hj,h^c=hj,i^b=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72255996, #72256017, #72256094, … |
| maximal normal subgroup of | #72255996, #72256017, #72256094, … |
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, 3, F.4*F.5*F.7);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.6*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);