Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #6346637 |
| commutator, Frattini, cosocle | C2×M16 | C25 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
4479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,8)(13,25)(16,30)(21,32);(1,2,9,29,24,25,5,19,7,8,27,20,6,13,23,12)(3,14,32,28,26,22,16,15,11,31,21,10,17,18,30,4);(1,2)(3,15,11,4)(5,20,23,29)(6,25)(7,8)(9,12,27,19)(10,26,28,17)(13,24)(14,30)(16,31)(18,32)(21,22);(1,3,6,26,7,11,24,17)(2,10,25,15,8,28,13,4)(5,21,9,30,23,32,27,16)(12,18,29,31,19,22,20,14);(1,4,7,15)(2,11,8,3)(5,22,23,18)(6,10,24,28)(9,14,27,31)(12,16,19,30)(13,26,25,17)(20,32,29,21)Polycyclic presentation:
2^10|b^2=fgi,c^2=fh,d^2=hi,e^2=j,g^2=ij,i^2=j,b^a=bf,c^a=cfhj,c^b=cgi,d^a=df,d^b=dfgh,d^c=dfij,e^a=eh,e^b=ej,e^c=eh,f^b=fh,f^c=fj,f^e=fj,g^a=gh,g^b=gh,g^c=ghij,g^d=gij,g^e=gj,g^f=gj,h^b=hj,h^c=hj,h^d=hj,i^c=ij,i^d=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72231730, #72250830, C2×#72231710, … |
| maximal normal subgroup of | #72231730, #72250830, C2×#72231710, … |
| maximal quotient of | #72250381, C2×#72231710, #72280113, … |
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.6*F.7*F.9);;
gap> SetPower(coll, 3, F.6*F.8);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);