Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C4×C8 | C25 |
| prime radical | #7624012 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17320 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,16)(13,15)(19,27)(20,28)(24,30)(25,32);(18,22)(19,20)(21,29)(24,32)(25,30)(26,31)(27,28);(1,3,5,11,6,12,4,13)(2,7,9,14,10,16,8,15)(17,18)(19,28,27,20)(21,31)(22,23)(24,25,30,32);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,16,15)(17,19,29,32,22,30,31,28,23,27,26,25,18,24,21,20);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,15,16,14)(17,20,31,30,18,25,29,19,23,28,21,24,22,32,26,27)Polycyclic presentation:
2^10|c^2=f,d^2=ghi,e^2=ghj,f^2=h,g^2=ij,i^2=j,c^b=cj,d^a=dj,d^b=dg,d^c=dghij,e^a=ehj,e^b=egi,e^c=eghj,e^d=ej,f^d=fj,f^e=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij109 conjugacy classes — view the full character table.
| maximal subgroup of | #72287030, #72287243, #72287589, … |
| maximal normal subgroup of | #72287030, #72287243, #72287589, … |
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, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*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);