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 | #2215 | C8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #11410 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
575 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)(3,8)(4,16)(5,10)(6,9)(7,11)(12,25)(13,21)(14,20)(15,22)(17,29)(18,23)(19,24)(26,32)(27,30)(28,31)(33,34)(35,37)(36,49)(38,44)(39,57)(40,42)(41,46)(43,53)(45,51)(47,55)(48,58)(50,54)(52,64)(56,61)(59,62)(60,63);(1,3,5,13,6,14,18,27,7,15,19,28,4,12,17,26)(2,8,10,21,9,20,23,30,11,22,24,31,16,25,29,32)(33,35,37,42,39,48,38,57,40,45,51,34,41,44,58,46)(36,50,52,61,54,49,63,60,55,62,56,64,59,47,53,43);(1,4,7,6)(2,9,11,16)(3,12,15,14)(5,17,19,18)(8,20,22,25)(10,23,24,29)(13,26,28,27)(21,30,31,32)(34,42)(35,45)(43,60)(44,48)(46,57)(47,49)(50,62)(61,64);(33,36,51,52,39,54,58,63,40,55,37,56,41,59,38,53)(34,43,44,62,46,61,35,47,42,60,48,50,57,64,45,49)Polycyclic presentation:
2^10|b^2=e,c^2=ij,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, #72321481, … |
| maximal normal subgroup of | #72320600, #72321126, #72321481, … |
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.9*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);