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 | #6437464 | C2 |
| LCS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
5335 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)(3,8,15,28)(4,9,18,31)(5,10,22,35)(6,24,25,11)(13,26,39,57)(14,40,42,27)(16,29,43,54)(17,34,21,30)(19,47,46,32)(20,48,50,33)(23,53,52,36)(37,55,41,58)(38,56,63,64)(44,59,49,61)(45,60,51,62);(1,3)(2,8)(4,13)(5,20,22,50)(6,16,25,43)(7,15)(9,26)(10,33,35,48)(11,29,24,54)(12,28)(14,23)(17,44,21,49)(18,39)(19,38,46,63)(27,36)(30,59,34,61)(31,57)(32,56,47,64)(37,45)(40,53)(41,51)(42,52)(55,60)(58,62);(1,4)(2,9)(3,13)(5,21)(6,19)(7,18)(8,26)(10,34)(11,32)(12,31)(14,41)(15,39)(16,38)(17,22)(20,49)(23,51)(24,47)(25,46)(27,58)(28,57)(29,56)(30,35)(33,61)(36,62)(37,42)(40,55)(43,63)(44,50)(45,52)(48,59)(53,60)(54,64);(1,5)(2,10)(3,14)(4,17,18,21)(6,23)(7,22)(8,27)(9,30,31,34)(11,36)(12,35)(13,37,39,41)(15,42)(16,20)(19,45,46,51)(24,53)(25,52)(26,55,57,58)(28,40)(29,33)(32,60,47,62)(38,44,63,49)(43,50)(48,54)(56,59,64,61)Polycyclic presentation:
2^10|a^2=j,b^2=h,d^2=eij,b^a=bh,c^a=cf,d^a=de,d^b=dg,d^c=df,e^b=ej,e^c=ej,f^d=fj,g^a=gi,g^b=gi,h^d=hi,h^g=hj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72303157, #72303348, #72303420 |
| maximal normal subgroup of | #72303157, #72303348, #72303420 |
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> SetPower(coll, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, 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, 4, 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);