Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #12067 |
| commutator | C16 | C22×C32 |
| Frattini, cosocle | #7052 | C23 |
| socle | C22 | #122963 |
| prime radical | C2×D32 | C16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ #93667 ◁ G | |
| LECS | G ▷ #7052 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
252 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,9,5,10,17,25,6,11,18,16,20,27,32,31,7,12,19,26,21,28,13,23,22,29,30,3,8,14,24,15)(34,39,41,50,42,51,54,53)(35,43,45,58)(36,46)(37,48)(40,49,55,52,56,62,64,63)(44,57,60,59)(47,61);(1,3,13,12,20,25,4,14,22,26,32,11,5,15,30,28,7,16,17,2,8,23,19,27,6,9,24,29,21,31,18,10)(34,40,54,64,42,56,41,55)(35,44,45,60)(36,47)(39,49,53,63,51,62,50,52)(43,57,58,59)(46,61);(33,34,35,39,36,40,43,52,37,41,44,53,46,55,59,49,38,42,45,51,47,56,58,63,48,54,60,50,61,64,57,62)Polycyclic presentation:
2^11|a^2=d,b^2=dfj,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,c^a=ce,c^b=cg,d^c=dik,e^a=eg,e^b=ei,e^d=ek,f^c=fk,g^a=gi,g^b=gk,i^a=ik188 conjugacy classes — view the full character table.
| maximal subgroup of | #72230720 |
| maximal normal subgroup of | #72230720 |
| maximal quotient of | #72230720, #72230759 |
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.4*F.6*F.10);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);