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×#13674 |
| 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 | |
1374 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 1374 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)(3,13)(4,19,30,61,22,54,23,38)(5,10)(6,12)(7,11)(8,14)(9,21,43,57,35,58,36,25)(15,48,52,53,46,63,20,55)(16,45)(17,33)(18,42)(24,59)(26,39)(27,60)(28,40)(29,41)(31,62,34,50,44,49,64,56)(32,47)(37,51);(1,3)(2,8)(4,20,23,46,22,52,30,15)(5,24)(6,17)(7,14)(9,34,36,31,35,64,43,44)(10,37)(11,13)(12,42)(16,28)(18,26)(19,53,38,48,54,55,61,63)(21,56,25,49,58,50,57,62)(27,47)(29,51)(32,40)(33,39)(41,59)(45,60);(3,14)(8,13)(15,46)(16,47)(17,18)(20,52)(24,51)(31,44)(32,45)(33,42)(34,64)(37,59)(48,63)(49,62)(50,56)(53,55);(1,4)(2,9)(3,15)(5,25)(6,30)(7,22)(8,31)(10,38)(11,35)(12,43)(13,44)(14,46)(16,50)(17,20)(18,52)(19,40)(21,28)(23,26)(24,49)(27,58)(29,57)(32,53)(33,34)(36,39)(37,63)(41,61)(42,64)(45,55)(47,56)(48,59)(51,62)(54,60)Polycyclic presentation:
2^10|a^2=ghi,b^2=ghj,e^2=hj,g^2=i,h^2=j,b^a=bj,c^b=cj,d^a=de,d^b=df,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=ij58 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.9);;
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, 2, 1, F.2*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, 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);