Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139764 |
| commutator | C2×#127 | C22×C4 |
| Frattini, prime radical, cosocle | C2×#535 | C23 |
| LCS | G ▷ C2×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
888 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 888 conjugacy classes of subgroups — too many to draw. Only its 69 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 — 3 generators on 64 points:
(1,2)(3,8)(4,18,21,9)(5,10)(6,12)(7,27)(11,26)(13,39,43,57)(14,29)(15,45)(16,30)(17,31)(19,33,48,47)(20,35,22,34)(23,36)(24,53)(25,37)(28,41,32,42)(38,51)(40,62,52,61)(44,56)(46,55)(49,59,50,60)(54,63,58,64);(1,3)(2,8)(4,13,20,42)(5,23)(6,17)(7,16,26,15)(9,32,34,57)(10,36)(11,30,27,45)(12,31)(14,25)(18,28,35,39)(19,40,49,63)(21,43,22,41)(24,46,51,44)(29,37)(33,58,59,62)(38,55,53,56)(47,54,60,61)(48,52,50,64);(1,4,5,19,6,21,25,48)(2,9,10,33,12,18,37,47)(3,13,14,40,17,43,23,52)(7,20,24,49,26,22,51,50)(8,28,29,54,31,32,36,58)(11,34,38,59,27,35,53,60)(15,42,46,63,16,41,44,64)(30,57,56,62,45,39,55,61)Polycyclic presentation:
2^10|a^2=g,b^2=f,c^2=d,d^2=j,h^2=j,b^a=bf,c^a=ce,c^b=ceghi,d^a=dg,d^b=dgj,e^a=ej,e^b=eghj,e^c=eg,e^d=ej,f^c=fghij,f^e=fij,g^c=gj,h^a=hi,h^b=hi,h^c=hj,h^e=hj,h^f=hj,i^b=ij,i^c=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72318432, #72318472, #72318854, … |
| maximal normal subgroup of | #72318432, #72318472, #72318854, … |
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.7);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);