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 | #119495 |
| 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 | |
2183 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)(3,11,13,7)(4,15,17,26)(5,10)(6,9)(8,18,25,20)(12,30)(14,29,19,23)(16,22)(21,27)(24,31)(28,32);(1,3)(2,7)(4,16)(5,19,21,14)(6,13)(8,24,25,30)(9,11)(10,23,27,29)(12,20,31,18)(15,28)(17,32)(22,26);(1,4)(2,8,9,25)(3,12)(5,20,21,18)(6,17)(7,22,11,28)(10,26)(13,31)(14,32,19,16)(15,27)(23,30)(24,29)Polycyclic presentation:
2^10|a^2=e,b^2=dehi,c^2=dfgj,f^2=j,g^2=j,b^a=bd,c^a=cf,c^b=cef,d^a=dj,d^c=dh,e^b=ej,e^c=ej,f^b=fh,f^c=fi,f^d=fj,g^a=gi,g^b=ghi,g^c=ghi,g^d=gj,h^a=hj,h^b=hj,i^c=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72255996, #72256054, #72256091, … |
| maximal normal subgroup of | #72255996, #72256054, #72256091, … |
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.4*F.5*F.8*F.9);;
gap> SetPower(coll, 3, F.4*F.6*F.7*F.10);;
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.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);