Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #3034720 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | #454367 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3631 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)(4,8,15,22)(5,9)(6,10)(11,20,27,17)(12,23)(13,19)(14,21,30,24)(16,25)(18,28)(26,31,29,32);(1,3)(2,7)(4,11)(5,12)(6,13)(8,20)(9,23)(10,19)(14,29)(15,27)(16,28)(17,22)(18,25)(21,32)(24,31)(26,30);(12,28)(18,23)(26,29)(31,32);(1,4)(2,8,10,22)(3,11)(5,14)(6,15)(7,17,19,20)(9,24,25,21)(12,26)(13,27)(16,30)(18,32,23,31)(28,29);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,21)(10,25)(11,26)(13,28)(15,30)(17,31)(19,23)(20,32)(22,24)(27,29)Polycyclic presentation:
2^10|a^2=g,d^2=i,c^b=cf,d^a=dfgh,d^b=dfi,e^a=ehi,e^b=egi,e^c=ehi,e^d=ei,f^e=fj,g^d=gj,h^a=hj,h^b=hj,i^a=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72318234, #72319027, #72319065, … |
| maximal normal subgroup of | #72318234, #72319027, #72319065, … |
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.7);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*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);