Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #93064 |
| commutator | C8×C16 | C42 |
| Frattini, cosocle | C2×C162 | C22 |
| prime radical | C23 | #7096 |
| LCS | G ▷ C8×C16 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×C162 ◁ G | |
| LECS | G ▷ C2×C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
376 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 512 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2)(3,7)(4,17)(5,9)(6,20)(8,14)(10,13)(11,18)(12,25)(15,21)(16,27)(19,32)(22,29)(23,31)(24,30)(26,28)(33,34,35,41,46,53,60,63,51,42,59,57,50,39,45,52,38,44,49,58,36,40,47,55,37,43,48,56,62,54,61,64);(2,7,9,21)(3,12)(4,13)(6,18)(8,24,25,10)(11,27,20,23)(14,15)(16,29)(17,22,32,26)(19,30)(28,31)(34,39)(35,45)(36,50)(37,51)(40,43)(41,56)(42,53)(44,54)(46,62)(47,48)(49,61)(55,63)(57,58)(59,60)Polycyclic presentation:
2^11|a^2=d,b^2=e,c^2=fghjk,d^2=gj,f^2=hj,g^2=ik,h^2=j,i^2=k,b^a=bc,c^a=cf,c^b=cfgi,d^b=dgijk,e^a=ejk,f^a=fh,f^b=fh,g^b=gi,h^a=hj,h^b=hj,i^b=ik182 conjugacy classes — view the full character table.
| maximal subgroup of | #72227058, #72227126, #72227135 |
| maximal normal subgroup of | #72227058, #72227126, #72227135 |
| maximal quotient of | #72227046 |
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.5);;
gap> SetPower(coll, 3, F.6*F.7*F.8*F.10*F.11);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);