Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #201687 |
| commutator | C25 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | #62842 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
3408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,8,16,29)(4,9,19,32)(5,10,23,36)(6,11,25,38)(13,26,41,52)(14,33,20,27)(15,37,24,28)(17,34,21,30)(18,31,46,57)(22,35,49,60)(39,50,42,53)(40,56,45,51)(43,54,47,58)(44,59,48,55)(61,64,62,63);(1,3)(2,8)(4,13)(5,20,23,14)(6,24,25,15)(7,16)(9,26)(10,33,36,27)(11,37,38,28)(12,29)(17,39,21,42)(18,45,46,40)(19,41)(22,43)(30,50,34,53)(31,56,57,51)(32,52)(35,54)(44,62)(47,49)(48,61)(55,64)(58,60)(59,63);(1,4)(2,9)(3,13)(5,21)(6,18)(7,19)(8,26)(10,34)(11,31)(12,32)(14,42)(15,40)(16,41)(17,23)(20,39)(22,48)(24,45)(25,46)(27,53)(28,51)(29,52)(30,36)(33,50)(35,59)(37,56)(38,57)(43,62)(44,49)(47,61)(54,64)(55,60)(58,63);(1,5,6,22)(2,10,11,35)(3,14,15,43)(4,17,18,44)(7,23,25,49)(8,27,28,54)(9,30,31,55)(12,36,38,60)(13,39,40,61)(16,20,24,47)(19,21,46,48)(26,50,51,63)(29,33,37,58)(32,34,57,59)(41,42,45,62)(52,53,56,64)Polycyclic presentation:
2^10|a^2=j,b^2=g,d^2=e,b^a=bg,c^a=ch,c^b=ch,d^a=df,d^b=df,d^c=dj,e^a=ei,e^b=ei,f^b=fh,f^c=fj,f^d=fi,g^d=gh,g^e=gj,h^d=hj,i^b=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72310871 |
| maximal normal subgroup of | #72310871 |
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);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*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, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
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);