Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13673 |
| commutator, Frattini, cosocle | #492 | C24 |
| LCS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1391 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 1391 conjugacy classes of subgroups — too many to draw. Only its 115 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 — 4 generators on 64 points:
(1,2)(3,8)(4,18,22,39,21,53,30,58)(5,23)(6,12)(7,11)(9,20,37,25,36,56,43,55)(10,29)(13,31)(14,46,19,52,44,63,51,62)(15,48)(16,42)(17,34)(24,57)(26,40)(27,41)(28,59)(32,47,61,49,60,64,35,54)(33,45)(38,50);(1,3)(2,8)(4,19,22,44,21,51,30,14)(5,24)(6,16)(7,13)(9,35,37,32,36,61,43,60)(10,38)(11,31)(12,42)(15,28)(17,26)(18,52,39,63,53,62,58,46)(20,54,25,47,56,49,55,64)(23,57)(27,45)(29,50)(33,41)(34,40)(48,59);(3,13)(8,31)(14,44)(15,45)(16,17)(19,51)(24,50)(32,60)(33,48)(34,42)(35,61)(38,57)(46,63)(47,64)(49,54)(52,62);(1,4)(2,9)(3,14)(5,25)(6,30)(7,21)(8,32)(10,39)(11,36)(12,43)(13,44)(15,49)(16,19)(17,51)(18,41)(20,28)(22,26)(23,58)(24,47)(27,56)(29,55)(31,60)(33,52)(34,35)(37,40)(38,46)(42,61)(45,54)(48,62)(50,64)(53,59)(57,63)Polycyclic presentation:
2^10|a^2=ghj,b^2=ghj,e^2=hj,g^2=i,h^2=j,c^b=cj,d^a=de,d^b=df,e^a=ej,e^b=eg,e^d=eh,f^a=fgh,f^b=fh,f^e=fh,g^b=gi,g^d=gh,g^f=gi,h^b=hi,h^d=hj,h^f=hj,i^d=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72306599 |
| maximal normal subgroup of | #72306599 |
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.8*F.10);;
gap> SetPower(coll, 2, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
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, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);