Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #149761 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | #1781 | C23 |
| prime radical | C22×H(5,𝔽2) | C2×C4 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
1011 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,7,12)(3,13,17,41)(4,18)(5,10,24,36)(6,11)(8,25)(9,22)(14,39)(15,45)(16,31)(19,50)(20,33)(21,37,26,34)(23,54)(27,35)(28,38)(29,48,32,44)(30,55,57,46)(40,47)(42,49)(43,61,51,63)(52,60,53,59)(56,62)(58,64);(1,3,5,15)(2,8,10,30)(4,19,20,51)(6,25,27,55,28,16,23,46)(7,17,24,47)(9,32,33,58)(11,13,35,40,38,41,54,45)(12,31,36,57)(14,42,43,22)(18,29,49,56)(21,48,53,64,26,44,52,62)(34,50,59,63,37,39,60,61);(1,4,7,22)(2,9,12,18)(3,14,17,19)(5,20,24,42)(6,26,28,21)(8,29,31,32)(10,33,36,49)(11,37,38,34)(13,39,41,50)(15,43,47,51)(16,48,25,44)(23,53,27,52)(30,56,57,58)(35,60,54,59)(40,61,45,63)(46,64,55,62)Polycyclic presentation:
2^10|a^2=fgij,b^2=d,c^2=j,d^2=g,b^a=be,c^a=ch,c^b=cgj,d^a=df,e^a=ehj,e^b=ef,e^c=ej,e^d=eh,f^a=fi,f^b=fh,f^c=fj,f^d=fi,g^a=gi,g^e=gj,h^a=hj,h^b=hi,h^d=hj,i^b=ij40 conjugacy classes — view the full character table.
| maximal subgroup of | #72311026, #72311034, #72311168, … |
| maximal normal subgroup of | #72311026, #72311034, #72311168, … |
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*F.7*F.9*F.10);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
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.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);