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 | #1683394 |
| 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 | |
2656 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)(3,11,12,7)(4,15,18,8)(5,9)(6,10)(13,21)(14,22,28,24)(16,27)(17,25,19,23)(20,26)(29,32)(30,31);(3,12)(13,16)(14,28)(17,19)(23,25)(31,32);(1,3)(2,7)(4,16)(5,14,20,28)(6,12)(8,21)(9,24,26,22)(10,11)(13,18)(15,27)(17,30,19,29)(23,32,25,31);(1,4,6,18)(2,8)(3,13,12,16)(5,19,20,17)(7,21)(9,25)(10,15)(11,27)(14,29,28,30)(22,31)(23,26)(24,32);(1,5)(2,9)(3,14)(4,17)(6,20)(7,22)(8,23)(10,26)(11,24)(12,28)(13,29)(15,25)(16,30)(18,19)(21,31)(27,32)Polycyclic presentation:
2^10|a^2=g,c^2=f,d^2=ij,b^a=bh,c^a=cghi,c^b=cij,d^a=dghij,d^b=df,d^c=dfij,e^a=eh,e^b=egh,e^c=ehij,e^d=ehj,f^e=fj,g^c=gj,g^d=gj,h^c=hj,i^a=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72334879, #72334887, #72334896, … |
| maximal normal subgroup of | #72334879, #72334887, #72334896, … |
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.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
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.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
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);