Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #150393 |
| commutator | #379 | C22×C4 |
| Frattini, cosocle | #1781 | C23 |
| prime radical | C22×H(5,𝔽2) | D4 |
| LCS | G ▷ #379 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
1096 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(1,2,5,10)(3,13,15,29)(4,9,18,32)(6,24,22,37)(7,12,23,38)(8,26,28,53)(11,39,40,25)(14,41,43,57)(16,35,47,30)(17,42,48,54)(19,50,49,60)(20,34,21,36)(27,44,56,63)(31,55,58,64)(33,52,61,51)(45,62,46,59);(1,3,7,17)(2,8)(4,14,20,45)(5,15)(6,16,25,26)(9,31)(10,35,38,28)(11,29,24,54)(12,30)(13,37)(18,46)(19,44,51,55)(21,43)(22,53)(23,48)(27,34)(32,58,36,56)(33,59,50,57)(39,47)(40,42)(41,60)(49,63)(52,64)(61,62);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,27)(10,36)(11,33)(12,34)(13,41)(15,46)(16,44)(17,45)(18,23)(22,52)(24,50)(25,51)(26,55)(28,58)(29,59)(30,31)(32,38)(35,56)(37,61)(39,49)(40,60)(42,62)(43,48)(47,64)(53,63)(54,57)Polycyclic presentation:
2^10|a^2=d,b^2=fgij,e^2=gh,b^a=bdegh,c^a=cij,c^b=cfgi,d^b=df,d^c=dj,e^a=efg,e^b=efgh,e^c=eh,e^d=ehi,f^a=fh,f^b=fj,f^c=fj,f^e=fij,g^a=gi,g^b=gj,g^c=gj,g^d=gj,g^e=gj,h^b=hi,h^e=hj,i^a=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72284403, #72284674, #72284743, … |
| maximal normal subgroup of | #72284403, #72284674, #72284743, … |
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.4);;
gap> SetPower(coll, 2, F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*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.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);