Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #151391 |
| commutator, Frattini, cosocle | #1781 | C23 |
| prime radical | C22×#120 | D4 |
| LCS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
641 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 641 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,14,25)(4,15)(5,19,21,9)(6,10)(7,13,24,16)(8,18)(12,29)(17,26,20,27)(22,32)(23,30)(28,31);(1,3,5,12,6,14,21,31)(2,7,9,23,10,24,19,22)(4,16,17,32,18,13,20,30)(8,25,26,28,15,11,27,29);(3,12)(7,22)(8,15)(11,28)(13,32)(14,31)(16,30)(23,24)(25,29)(26,27)Polycyclic presentation:
2^10|a^2=fi,b^2=e,d^2=fij,e^2=j,b^a=bd,c^a=cghi,c^b=cefhij,d^a=df,d^b=dfg,d^c=dgij,e^a=eg,e^c=eij,e^d=ehi,f^b=fi,f^c=fij,f^e=fj,g^a=gj,g^b=gh,g^d=gij,h^a=hi,h^d=hj,i^b=ij28 conjugacy classes — view the full character table.
| maximal subgroup of | #72227531, #72227537, #72338673, … |
| maximal normal subgroup of | #72227531, #72338673, #72338793, … |
| maximal quotient of | #72227531 |
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.6*F.9);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 4, F.6*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);