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 | #2140339 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | C2×#2021 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3034 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)(3,11,14,7)(4,15,16,8)(5,9)(6,10)(12,27)(13,24,18,22)(17,25,19,23)(20,26)(21,28)(29,32)(30,31);(4,16)(12,28)(23,25)(31,32);(1,3)(2,7)(4,12)(5,18,20,13)(6,14)(8,21)(9,24,26,22)(10,11)(15,27)(16,28)(17,30,19,29)(23,32,25,31);(1,4,6,16)(2,8)(3,12,14,28)(5,19,20,17)(7,21)(9,25)(10,15)(11,27)(13,29,18,30)(22,31)(23,26)(24,32);(1,5)(2,9)(3,13)(4,17)(6,20)(7,22)(8,23)(10,26)(11,24)(12,29)(14,18)(15,25)(16,19)(21,31)(27,32)(28,30)Polycyclic presentation:
2^10|a^2=g,c^2=f,d^2=ij,b^a=bgh,c^a=cghi,d^a=dgij,d^b=dfij,d^c=df,e^b=egh,e^c=eh,e^d=ehj,f^e=fj,g^c=gj,g^d=gj,h^c=hj,i^a=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72334827, #72334833, #72334879, … |
| maximal normal subgroup of | #72334827, #72334833, #72334879, … |
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, 3, F.6);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);