Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #7155 |
| commutator | C2×C16 | C4×C16 |
| Frattini, cosocle | C8×M64 | C22 |
| socle | C2 | #72225409 |
| prime radical | C23 | #7456 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #1397 ◁ C4×M64 ◁ C8×M64 ◁ G | |
| LECS | G ▷ C8×M64 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,12,5,15,23,46,7,17,25,48,4,27,22,58,9,18,20,49,30,54,56,62,29,13,51,52,32,11,38,39,10,19,21,50,33,14,53,45,36,16,55,47,28,41,57,43,8,37,26,61,34,40,60,64,6,35,24,44,31,42,59,63);(2,11,13,40,37,27,17,14,19,42,35,54,18,41,16,15)(3,20,21,26)(4,28)(5,33)(12,43,44,45,61,39,47,62,50,58,52,46,49,63,48,64)(22,59,57,38)(23,60,53,56)(24,25,51,55)(30,34)(31,32)Polycyclic presentation:
2^11|a^2=c,b^2=d,c^2=f,d^2=gh,e^2=gk,f^2=h,g^2=i,h^2=j,i^2=k,j^2=k,b^a=be,c^b=cj,d^a=dg,d^c=dk,e^a=egijk,e^b=ek,f^b=fk,g^a=gi,i^a=ik224 conjugacy classes — view the full character table.
| maximal subgroup of | #72225417, #72225515, #72225534 |
| maximal normal subgroup of | #72225417, #72225515, #72225534 |
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.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 5, F.7*F.11);;
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> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);