Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152049 |
| commutator | C4×M16 | C22×C4 |
| Frattini, cosocle | #3151 | C23 |
| prime radical | #60801 | C22 |
| LCS | G ▷ C4×M16 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C22×C42 ◁ #60801 ◁ G | |
| LECS | G ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1092 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 1092 conjugacy classes of subgroups — too many to draw. Only its 81 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)(3,13)(4,18)(5,11)(6,10)(7,12)(8,15)(9,20)(14,28)(16,36)(17,41)(19,48)(21,32)(22,49)(23,29)(24,37)(25,34)(26,39)(27,35)(30,44)(31,45)(33,53)(38,54)(40,42)(43,57)(46,58)(47,59)(50,60)(51,61)(52,64)(55,63)(56,62);(1,3)(2,8)(4,19,22,52,20,45,53,50)(5,23)(6,15)(7,17)(9,31,33,60,18,48,49,64)(10,13)(11,36)(12,30)(14,21,43,55,42,24,51,56)(16,25)(26,46)(27,44)(28,32,57,63,40,37,61,62)(29,34)(35,41)(38,58)(39,59)(47,54);(1,4)(2,9,10,18)(3,14,15,42)(5,24,26,56)(6,20)(7,22,27,53)(8,28)(11,37,38,63)(12,33)(13,40)(16,45,46,52)(17,43)(19,47,50,23)(21,54,55,25)(29,48,58,64)(30,57,41,61)(31,59,60,36)(32,39,62,34)(35,49)(44,51)Polycyclic presentation:
2^10|b^2=gi,c^2=e,f^2=hj,g^2=ij,h^2=j,b^a=bi,c^a=cd,c^b=cf,d^b=dij,d^c=dj,e^a=ej,e^b=eg,f^a=fij,f^b=fj,f^c=fgh,f^d=fj,f^e=fh,g^c=gh,g^e=gij,h^c=hi,h^e=hj,i^c=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72306785, #72306869, #72306923, … |
| maximal normal subgroup of | #72306785, #72306869, #72306923, … |
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, 2, F.7*F.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);