Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #394267 |
| commutator, Frattini, cosocle | C23×D4 | C24 |
| prime radical | #6437629 | C2 |
| LCS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
5351 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)(3,8,15,27)(4,9,19,31)(5,10,21,33)(6,11,24,36)(13,37,39,25)(14,26,41,52)(16,35,23,28)(17,29,43,54)(18,44,45,30)(20,32,47,56)(22,34,48,57)(38,50,60,59)(40,58,49,51)(42,61,62,53)(46,63,64,55);(1,3)(2,8)(4,17)(5,14)(6,23,24,16)(7,15)(9,29)(10,26)(11,35,36,28)(12,27)(13,20,39,47)(18,42)(19,43)(21,41)(22,49,48,40)(25,32,37,56)(30,53)(31,54)(33,52)(34,58,57,51)(38,46,60,64)(44,61)(45,62)(50,55,59,63);(1,4)(2,9)(3,13)(5,18)(6,20)(7,19)(8,25)(10,30)(11,32)(12,31)(14,38)(15,39)(16,17)(21,45)(22,46)(23,43)(24,47)(26,50)(27,37)(28,29)(33,44)(34,55)(35,54)(36,56)(40,42)(41,60)(48,64)(49,62)(51,53)(52,59)(57,63)(58,61);(1,5)(2,10)(3,14)(4,18)(6,22)(7,21)(8,26)(9,30)(11,34)(12,33)(13,38)(15,41)(16,40)(17,42)(19,45)(20,46)(23,49)(24,48)(25,50)(27,52)(28,51)(29,53)(31,44)(32,55)(35,58)(36,57)(37,59)(39,60)(43,62)(47,64)(54,61)(56,63)Polycyclic presentation:
2^10|a^2=j,b^2=h,b^a=bh,c^a=ce,c^b=cg,d^a=df,e^b=ej,e^d=ej,f^c=fj,g^a=gi,g^b=gi,h^c=hi,h^g=hj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72303157, #72303348, #72303400 |
| maximal normal subgroup of | #72303157, #72303348, #72303400 |
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.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, 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, 7, 2, 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, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);