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)(3,14,19,44)(4,20)(5,10)(6,12)(7,11)(8,16,33,51)(9,23)(13,28)(15,45,50,42)(17,37,18,43)(21,55,56,62)(22,54)(24,36)(25,31,52,32)(26,39)(27,41)(29,38)(30,57,63,47)(34,48,53,49)(35,46)(40,59)(58,61)(60,64);(1,3)(2,8,11,31,12,33,13,32)(4,21,23,56)(5,25,26,52)(6,19)(7,18)(9,34,36,42,20,53,40,45)(10,37,38,14,39,43,41,44)(15,46)(16,27,51,29)(17,28)(22,50)(24,57,59,47)(30,61,62,35,63,64,55,54)(48,58)(49,60);(1,4)(2,9)(3,15,19,50)(5,22)(6,23)(7,24)(8,30,33,63)(10,35)(11,40)(12,20)(13,36)(14,42,44,45)(16,47,51,57)(17,48,18,49)(21,52,56,25)(26,46)(27,58)(28,59)(29,60)(31,62,32,55)(34,43,53,37)(38,61)(39,54)(41,64)Polycyclic presentation:
2^10|a^2=gj,b^2=e,c^2=gj,d^2=j,e^2=i,h^2=j,b^a=bd,c^a=cj,c^b=cdf,d^b=dh,e^a=eh,e^c=eg,e^d=eij,f^b=fghj,f^e=fi,g^b=gj,h^a=hj,h^b=hi,h^d=hj,h^f=hj,i^a=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72319632, #72319665, #72319711, … |
| maximal normal subgroup of | #72319632, #72319665, #72319711, … |
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*F.10);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9);;
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, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, 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, 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, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);