Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13346 |
| commutator | #532 | C22×C4 |
| Frattini, cosocle | C2×#532 | C23 |
| socle | C2 | #147749 |
| prime radical | C2×#33546 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,5)(3,8,12,20)(4,9,19,16)(6,21,22,10)(11,15,28,31)(13,24,30,14)(17,26,18,27)(23,25,32,29);(1,3)(2,8)(4,15,17,23)(5,20)(6,14,22,24)(7,12)(9,25,26,28)(10,13,21,30)(11,16,29,27)(18,32,19,31);(1,4)(2,9)(3,11,12,28)(5,16)(6,17)(7,19)(8,23,20,32)(10,26)(13,15,30,31)(14,29,24,25)(18,22)(21,27)Polycyclic presentation:
2^10|a^2=j,b^2=g,c^2=d,e^2=j,h^2=j,b^a=bg,c^a=cf,c^b=ce,d^b=dj,e^b=eh,f^b=fhj,g^c=gh,g^e=gij,g^f=gij,h^a=hi,h^b=hi,h^c=hj,h^e=hj,h^f=hj,h^g=hj,i^b=ij,i^c=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72284626, #72286463, #72286762, … |
| maximal normal subgroup of | #72284626, #72286463, #72286762, … |
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.10);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, 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);