Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×#342 |
| commutator | C22×C4 | C43 |
| Frattini, prime radical, cosocle | C23×#41 | C23 |
| socle | C2 | #123452 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ C2×#3428 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C22 ▷ C2 ▷ C1 | |
4091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,8)(4,17)(5,10)(6,24,25,11)(7,12)(9,21,32,49)(13,35)(14,27)(15,28)(16,41,40,29)(18,42)(19,43)(20,44)(22,50,51,33)(23,34)(26,37,52,58)(30,46,55,63)(31,48,56,64)(36,57)(38,53)(39,60,59,54)(45,61)(47,62);(1,3)(2,8)(4,18,20,45)(5,14)(6,16)(7,15)(9,30,32,55)(10,27)(11,29)(12,28)(13,19,36,47)(17,42,44,61)(21,46,49,63)(22,39)(23,38)(24,41)(25,40)(26,31,52,56)(33,54)(34,53)(35,43,57,62)(37,48,58,64)(50,60)(51,59);(1,4)(2,9)(3,13,15,36)(5,19)(6,21)(7,20)(8,26,28,52)(10,31)(11,17)(12,32)(14,18,38,45)(16,37,40,58)(22,48)(23,47)(24,44)(25,49)(27,30,53,55)(29,35,41,57)(33,43)(34,56)(39,46,59,63)(42,60,61,54)(50,62)(51,64)Polycyclic presentation:
2^10|a^2=h,b^2=f,c^2=e,d^2=j,c^a=cg,c^b=cd,e^b=ej,f^c=fj,g^a=gi,h^c=hi,h^g=hj,i^a=ij130 conjugacy classes — view the full character table.
| maximal subgroup of | #72280630, #72281120, #72281170, … |
| maximal normal subgroup of | #72280630, #72281120, #72281170, … |
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.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, 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);