Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13049 |
| commutator | #127 | C23×C4 |
| Frattini, cosocle | C2×#127 | C24 |
| prime radical | #6348449 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,18)(5,11)(6,12)(7,14)(8,13)(10,21)(15,30)(16,25)(17,26)(19,27)(20,29)(22,32)(23,28)(24,31);(1,3)(2,9)(4,19,20,15,21,22,23,31)(5,7)(6,16)(8,17)(10,27,28,30,18,32,29,24)(11,14)(12,25)(13,26);(1,4,6,20,8,21,7,23)(2,10,12,28,13,18,14,29)(3,15,16,19,17,31,5,22)(9,24,25,32,26,30,11,27);(1,5,7,3,8,16,6,17)(2,11,14,9,13,25,12,26)(4,15,20,22,21,31,23,19)(10,24,28,27,18,30,29,32)Polycyclic presentation:
2^10|b^2=g,c^2=e,d^2=h,e^2=j,g^2=ij,h^2=j,b^a=bij,c^a=cj,c^b=cf,d^b=dghi,d^c=dhi,e^b=eij,e^d=ej,f^a=fj,f^b=fh,f^c=fij,f^d=fhj,f^e=fj,g^c=ghj,g^f=gij,h^b=hi,h^c=hj,h^f=hj,i^c=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72306595, #72306649, #72306757, … |
| maximal normal subgroup of | #72306595, #72306649, #72306757, … |
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, 2, F.7);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, 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.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);