Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #133849 |
| commutator | C22×#41 | C22×C4 |
| Frattini, cosocle | C2×#535 | C23 |
| prime radical | C2×#3021 | C22 |
| LCS | G ▷ C22×#41 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35327 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
2511 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 2511 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:
(1,2,6,12)(3,13,19,44)(4,20,23,9)(5,25,14,10)(7,29,28,11)(8,31,34,16)(15,48,52,43)(17,40,18,45)(21,55,57,64)(22,37,47,54)(24,39,59,38)(26,42,46,35)(27,41,30,32)(33,49,62,56)(36,51,53,50)(58,61,60,63);(1,3,14,46)(2,8,32,44,12,34,41,13)(4,21,22,52)(5,26,6,19)(7,18,30,16)(9,36,63,64,20,53,61,55)(10,40,29,35,25,45,11,42)(15,23,57,47)(17,27,31,28)(24,56,58,50)(33,39,48,54,62,38,43,37)(49,60,51,59);(1,4)(2,9)(3,15,19,52)(5,22)(6,23)(7,24)(8,33,34,62)(10,37)(11,39)(12,20)(13,43,44,48)(14,47)(16,49,31,56)(17,50,18,51)(21,46,57,26)(25,54)(27,58)(28,59)(29,38)(30,60)(32,61)(35,55,42,64)(36,40,53,45)(41,63)Polycyclic presentation:
2^10|a^2=j,b^2=de,c^2=gj,d^2=j,h^2=j,b^a=bd,c^a=cj,c^b=cdf,d^a=dj,d^b=dh,e^a=ehj,e^b=eh,e^c=eg,e^d=eij,f^a=fj,f^b=fghj,f^e=fi,g^b=gj,h^a=hj,h^b=hi,h^d=hj,h^e=hj,h^f=hj,i^a=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72318267, #72318311, #72318432, … |
| maximal normal subgroup of | #72318267, #72318311, #72318432, … |
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.4*F.5);;
gap> SetPower(coll, 3, F.7*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, 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, 4, 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, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);