Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72237006 |
| commutator | C22×#120 | C22×C4 |
| Frattini, cosocle | C2×#2613 | C23 |
| prime radical | #72234108 | C2 |
| LCS | G ▷ C22×#120 ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2613 ◁ G | |
| LECS | G ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
3686 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 3686 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,2,4,8)(3,7,12,21,13,22,29,23)(5,9,15,20)(6,10,18,26)(11,27,28,32,17,19,30,31)(14,24,16,25);(3,11,13,17)(7,19,22,27)(12,28,29,30)(21,31,23,32);(1,3,4,12)(2,7,20,21,9,22,8,23)(5,13,15,29)(6,17,14,30)(10,19,24,32,25,27,26,31)(11,18,28,16)Polycyclic presentation:
2^11|a^2=d,b^2=i,c^2=dehik,d^2=i,f^2=k,b^a=bi,c^a=ce,c^b=cf,d^c=dh,e^a=eh,e^b=eg,e^c=ehj,e^d=ej,f^a=fgj,f^c=fgik,f^d=fik,f^e=fjk,g^a=gi,g^c=gij,h^a=hj,h^b=hi,h^c=hjk,h^d=hk,h^f=hk,i^c=ik,j^a=jk,j^c=jk53 conjugacy classes — view the full character table.
| maximal subgroup of | #72249355, #72262498, #72262537, … |
| maximal normal subgroup of | #72249355, #72262498, #72262537, … |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.4*F.5*F.8*F.9*F.11);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 6, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, 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, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);