Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #144412 |
| commutator, Frattini, cosocle | C2×#535 | C23 |
| prime radical | C2×#3019 | C22 |
| LCS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1551 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 1551 conjugacy classes of subgroups — too many to draw. Only its 61 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)(4,19,21,53)(5,12,27,10)(6,28)(7,13)(8,18)(9,22,38,59)(11,30,41,26)(15,45,20,55)(16,35,25,33)(17,40,49,62)(23,52,24,54)(29,34)(31,42)(32,58,61,64)(36,56,43,46)(37,57,39,60)(44,50)(47,63,48,51);(1,3)(2,8)(4,20)(5,25)(6,29)(7,18)(9,36)(10,35)(11,40)(12,33)(13,14)(15,21)(16,27)(17,26)(19,51)(22,58)(23,48)(24,47)(28,44)(30,49)(31,50)(32,37)(34,42)(38,43)(39,61)(41,62)(45,54)(46,60)(52,55)(53,63)(56,57)(59,64);(1,4,7,24)(2,9)(3,15)(5,21,27,23)(6,22,31,60)(8,32,14,43)(10,37)(11,19)(12,38)(13,39)(16,20)(17,46)(18,48)(25,47)(26,57,30,59)(28,52)(29,56)(33,36,35,61)(34,51,44,45)(40,55,62,63)(41,54)(42,53)(49,58)(50,64)Polycyclic presentation:
2^10|a^2=f,c^2=hj,d^2=j,e^2=j,g^2=j,b^a=bfj,c^a=ce,c^b=cd,d^a=dh,d^b=dj,e^a=eg,e^b=eghi,f^c=fg,f^d=fj,f^e=fij,g^a=gi,g^b=gj,g^c=gj,g^e=gj,g^f=gj,h^a=hj,h^b=hj,i^a=ij,i^b=ij,i^c=ij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72317996, #72318229, #72318948, … |
| maximal normal subgroup of | #72317996, #72318229, #72318948, … |
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.6);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);