Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#2127 |
| commutator | C42 | C22×C42 |
| Frattini, cosocle | #532 | C24 |
| socle | C2 | C2×#12582 |
| prime radical | C2×#3449 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C24×C4 ◁ C2×#3449 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,8)(4,17,21,48)(5,22,24,33)(6,26,27,11)(7,12)(9,20,36,53)(10,25)(13,39,41,61)(14,42,23,55)(15,32)(16,30,44,31)(18,45,50,62)(19,46,52,47)(28,40,59,63)(29,43)(34,51,35,54)(37,56)(38,57)(49,60,64,58);(1,3)(2,8)(4,18)(5,23)(6,16)(7,15)(9,28)(10,37)(11,31)(12,32)(13,19)(14,24)(17,45)(20,40)(21,50)(22,55)(25,56)(26,30)(27,44)(29,38)(33,42)(34,58)(35,60)(36,59)(39,46)(41,52)(43,57)(47,61)(48,62)(49,51)(53,63)(54,64);(1,4,5,20)(2,9,33,48)(3,13,23,49)(6,21,25,54)(7,19,24,51)(8,28,42,62)(10,17,11,35)(12,34,22,47)(14,40,15,18)(16,41,56,63)(26,36,38,46)(27,52,57,53)(29,39,30,59)(31,60,37,45)(32,58,55,61)(43,64,44,50);(4,19)(9,34)(13,18)(17,46)(20,51)(21,52)(28,58)(35,36)(39,45)(40,49)(41,50)(47,48)(53,54)(59,60)(61,62)(63,64)Polycyclic presentation:
2^10|a^2=g,c^2=efi,e^2=i,f^2=i,h^2=j,c^a=ce,c^b=cij,d^c=dj,e^a=eh,e^b=ej,f^a=fhi,f^b=fj,g^c=ghij,g^e=gij,g^f=gi,h^a=hi,h^g=hj,i^a=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72284626, #72286893, #72287241, … |
| maximal normal subgroup of | #72284626, #72286893, #72287241, … |
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);;
gap> SetPower(coll, 3, F.5*F.6*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);