Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #134037 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | #2308 | C23 |
| prime radical | #62842 | C4 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #35448 ◁ G | |
| LECS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
2389 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 2389 conjugacy classes of subgroups — too many to draw. Only its 95 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)(3,13)(4,18,21,9)(5,22,26,10)(6,11,28,40)(8,23)(14,43)(15,31)(16,49)(17,35)(19,33,24,36)(20,34,55,38)(25,39,27,37)(29,47)(30,48)(32,44)(41,46)(42,52)(45,62)(50,61)(51,57)(53,60,54,59)(56,64)(58,63);(1,3,6,16)(2,8,11,30)(4,14,20,45,21,46,55,50)(5,23,25,51)(7,17,28,52)(9,32,34,58,18,29,38,56)(10,35,37,49)(12,31,40,57)(13,39,42,22)(15,27,48,26)(19,44,53,63,24,47,54,64)(33,41,59,61,36,43,60,62);(1,4,7,21)(2,9)(3,14)(5,24,26,19)(6,20)(8,29,31,32)(10,36)(11,38,40,34)(12,18)(13,41,35,43)(15,47)(16,50,52,45)(17,46)(22,33)(23,44)(25,54)(27,53)(28,55)(30,56)(37,59,39,60)(42,61)(48,63,51,64)(49,62)(57,58)Polycyclic presentation:
2^10|a^2=ij,b^2=e,c^2=ghij,e^2=gi,b^a=bd,c^a=cgi,c^b=cf,d^a=dj,d^b=dg,d^c=dj,e^a=eg,e^c=eh,e^d=ej,f^b=fh,f^c=fi,f^d=fj,f^e=fi,g^b=gj,g^c=gj,h^a=hj,h^b=hi,h^c=hj,h^e=hj,i^b=ij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72284595, #72284615, #72284674, … |
| maximal normal subgroup of | #72284595, #72284615, #72284674, … |
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.9*F.10);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
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, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);