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 | #485081 |
| commutator, Frattini, cosocle | #379 | C24 |
| prime radical | C22×H(5,𝔽2) | D4 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 1580 conjugacy classes of subgroups — too many to draw. Only its 125 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)(3,13)(4,18,22,9)(5,23)(6,11,29,41)(8,30)(10,27)(14,44)(15,43,24,53)(16,33)(17,38)(19,42)(20,34)(21,39,28,35)(25,36)(26,37)(31,49)(32,48,52,54)(40,56)(45,62,47,61)(46,50)(51,60)(55,59)(57,63,58,64);(1,3,7,17)(2,8,12,33)(4,19)(5,24,27,15)(6,16)(9,31)(10,32)(11,38,41,13)(14,22)(18,50)(20,47)(21,49,28,46)(23,52)(25,45)(26,54)(29,30)(34,58,36,57)(35,42)(37,53)(39,44)(40,43)(48,56)(51,64,55,63)(59,62,60,61);(1,4)(2,9)(3,14)(5,25)(6,28)(7,22)(8,31)(10,36)(11,39)(12,18)(13,42)(15,47)(16,49)(17,19)(20,27)(21,29)(23,34)(24,45)(26,51)(30,46)(32,58)(33,50)(35,41)(37,59)(38,44)(40,60)(43,62)(48,63)(52,57)(53,61)(54,64)(55,56);(1,5,7,27)(2,10)(3,15)(4,20,22,25)(6,26)(8,32)(9,34)(11,40,41,37)(12,23)(13,43,38,53)(14,45)(16,48,30,54)(17,24)(18,36)(19,47)(21,51)(28,55)(29,56)(31,57)(33,52)(35,60,39,59)(42,61,44,62)(46,63,49,64)(50,58)Polycyclic presentation:
2^10|a^2=gj,b^2=f,d^2=ghij,e^2=gj,b^a=be,c^a=cj,c^b=cgi,d^a=dgij,d^b=dfhij,d^c=dj,e^a=eg,e^b=egh,e^c=ej,e^d=ehi,f^a=fh,f^c=fj,f^d=fi,f^e=fij,g^b=gj,g^d=gj,h^b=hi,h^d=hj,h^e=hj,i^a=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72340929, #72340946, #72341228, … |
| maximal normal subgroup of | #72340929, #72340946, #72341228, … |
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.10);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 4, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);