Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13348 |
| commutator, Frattini, cosocle | C2×#127 | C24 |
| prime radical | C2×#33557 | C2 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×C4 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,13,29)(4,16,18,27)(5,20)(6,9)(7,14,24,15)(8,21,26,19)(10,22)(12,25)(17,28)(23,30)(31,32);(1,3)(2,7,10,11,9,24,20,29)(4,17,19,12,18,31,21,30)(5,14)(6,13)(8,25)(15,22)(16,32)(23,26)(27,28);(4,18)(8,26)(12,30)(16,27)(17,31)(19,21)(23,25)(28,32);(1,4,6,18)(2,8,9,26)(3,12,13,30)(5,21,22,19)(7,23,24,25)(10,27,20,16)(11,28,29,32)(14,31,15,17)Polycyclic presentation:
2^10|a^2=e,b^2=g,d^2=j,g^2=ij,h^2=j,b^a=bf,d^a=dehi,d^b=dfgi,d^c=dj,e^b=ej,e^d=ej,f^a=fj,f^b=fh,f^d=fhj,g^a=ghj,g^d=ghj,g^f=gij,h^a=hj,h^b=hi,h^d=hj,h^f=hj,i^a=ij,i^d=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72306731, #72317520, #72317694, … |
| maximal normal subgroup of | #72306731, #72317520, #72317694, … |
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.5);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);