Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #144366 |
| commutator, Frattini, cosocle | C2×#535 | C23 |
| prime radical | #449491 | C2 |
| LCS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
2262 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 2262 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,7,12)(3,8,18,33)(4,9,22,37)(5,10,25,40)(6,27,13,11)(14,44,48,58)(15,30,17,32)(16,31,51,62)(19,52,50,63)(20,39,24,35)(21,36,43,56)(23,38,49,60)(26,42,28,41)(29,47,59,46)(34,54,61,53)(45,64,55,57);(1,3,13,17)(2,8,27,32)(4,19,22,50)(5,23,28,16)(6,15,7,18)(9,34,37,61)(10,38,42,31)(11,30,12,33)(14,45)(20,46,24,47)(21,54)(25,49,26,51)(29,57)(35,44,39,58)(36,63)(40,60,41,62)(43,53)(48,55)(52,56)(59,64);(1,4,7,22)(2,9,12,37)(3,14,18,48)(5,24,25,20)(6,21,13,43)(8,29,33,59)(10,39,40,35)(11,36,27,56)(15,46,17,47)(16,50,51,19)(23,54,49,53)(26,55,28,45)(30,44,32,58)(31,61,62,34)(38,63,60,52)(41,64,42,57)Polycyclic presentation:
2^10|a^2=j,b^2=fgi,c^2=j,e^2=j,g^2=j,b^a=bf,c^a=cd,c^b=ce,d^b=dg,e^a=eh,e^b=eghj,e^c=ej,e^d=ej,f^b=fi,f^c=fgh,f^d=fij,f^e=fij,g^a=gi,g^b=gij,g^d=gj,g^e=gj,g^f=gj,h^c=hj,i^b=ij,i^c=ij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72318132, #72318229, #72318311, … |
| maximal normal subgroup of | #72318132, #72318229, #72318311, … |
| maximal quotient of | #72339626 |
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.10);;
gap> SetPower(coll, 2, F.6*F.7*F.9);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, 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);