Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#442 |
| commutator | C2×C8 | C23×C8 |
| Frattini, cosocle | C82 | C24 |
| socle | C22 | C2×#1580 |
| prime radical | #33733 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #11410 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
611 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,7)(4,9)(5,10)(6,8)(11,18)(12,19)(13,17)(14,22)(15,20)(16,21)(23,29)(24,27)(25,28)(26,30)(31,32)(34,41)(35,39)(36,49)(37,48)(38,47)(42,57)(43,58)(44,55)(45,54)(46,62)(50,61)(51,59)(52,53)(63,64);(1,3,4,11,5,12,14,23,6,13,15,24,16,25,26,31)(2,7,9,18,10,19,22,29,8,17,20,27,21,28,30,32)(33,34,37,41,35,44,47,57,40,45,38,54,39,42,48,55)(36,50,49,63,46,43,59,60,52,58,53,64,51,61,62,56);(2,8)(7,17)(9,20)(10,21)(18,27)(19,28)(22,30)(29,32)(33,35,40,39)(34,42,45,44)(36,46,52,51)(37,47,38,48)(41,55,54,57)(43,50,61,58)(49,59,53,62)(56,64,60,63);(33,36,47,62,39,51,37,53,40,52,48,59,35,46,38,49)(34,43,55,64,42,50,54,60,45,61,57,63,44,58,41,56)Polycyclic presentation:
2^10|b^2=e,c^2=j,d^2=fh,e^2=g,f^2=hj,g^2=i,h^2=j,b^a=bf,c^a=ci,c^b=cj,d^a=df,d^b=dh,e^a=eh,e^d=ej,f^a=fh,f^b=fj,g^a=gj,h^a=hj136 conjugacy classes — view the full character table.
| maximal subgroup of | #72320600, #72321126, #72321513, … |
| maximal normal subgroup of | #72320600, #72321126, #72321513, … |
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, 2, F.5);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);