Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2072791 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | #429959 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,17)(6,10)(9,19)(11,20)(12,26)(13,22)(14,23)(15,24)(16,25)(18,21)(27,31)(28,32)(29,30);(1,3,6,13)(2,7,10,22)(4,14)(5,18,19,12)(8,23)(9,26,17,21)(11,16)(15,28)(20,25)(24,32)(27,29)(30,31);(1,4)(2,8)(3,11,13,14)(5,15)(6,16)(7,20,22,23)(9,24)(10,25)(12,27,18,28)(17,30)(19,29)(21,31,26,32);(1,5)(2,9,10,17)(3,12)(4,15)(6,19)(7,21,22,26)(8,24,25,30)(11,27)(13,18)(14,28)(16,29)(20,31,23,32);(7,22)(20,23)(21,26)(31,32)Polycyclic presentation:
2^10|b^2=f,c^2=h,d^2=i,c^a=cg,c^b=cfgh,d^a=dfgj,d^b=df,e^a=eh,e^b=ei,f^c=fj,g^d=gj,h^b=hj,i^a=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72319168, #72319473, #72334852, … |
| maximal normal subgroup of | #72319168, #72319473, #72334852, … |
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);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);