Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #508087 |
| commutator, Frattini, cosocle | C2×#127 | C24 |
| prime radical | C2×#3019 | C22 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1825 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 1825 conjugacy classes of subgroups — too many to draw. Only its 129 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:
(1,2,7,12)(3,8,15,28)(4,9,19,32)(5,10,22,35)(6,11,25,38)(13,39,41,26)(14,34,44,55)(16,29,17,30)(18,46,48,31)(20,33,51,61)(21,27,43,56)(23,37,24,36)(40,62,64,58)(42,57,45,54)(47,63,52,53)(49,59,50,60);(1,3)(2,8)(4,13,19,41)(5,21,23,14,22,43,24,44)(6,16)(7,15)(9,26,32,39)(10,34,36,27,35,55,37,56)(11,29)(12,28)(17,25)(18,47,49,40,48,52,50,64)(20,45,51,42)(30,38)(31,58,59,53,46,62,60,63)(33,57,61,54);(1,4)(2,9)(3,13)(5,18)(6,20)(7,19)(8,26)(10,31)(11,33)(12,32)(14,40)(15,41)(16,45)(17,42)(21,52)(22,48)(23,50)(24,49)(25,51)(27,53)(28,39)(29,57)(30,54)(34,62)(35,46)(36,60)(37,59)(38,61)(43,47)(44,64)(55,58)(56,63);(1,5,7,22)(2,10,12,35)(3,14)(4,18,19,48)(6,24,25,23)(8,27)(9,31,32,46)(11,37,38,36)(13,40)(15,44)(16,43)(17,21)(20,49,51,50)(26,53)(28,56)(29,55)(30,34)(33,59,61,60)(39,63)(41,64)(42,52)(45,47)(54,62)(57,58)Polycyclic presentation:
2^10|a^2=j,b^2=g,d^2=eij,g^2=i,h^2=j,b^a=bg,c^a=ce,c^b=cej,d^a=def,d^b=df,e^b=ej,e^d=ej,f^b=fh,f^c=fj,f^d=fj,g^a=gi,g^c=gj,g^d=ghj,g^f=gi,h^a=hi,h^b=hij,h^d=hj,h^f=hj,i^d=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72306877, #72306893, #72306908, … |
| maximal normal subgroup of | #72306877, #72306893, #72306908, … |
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.10);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.5*F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
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);