Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #148234 |
| commutator | C2×#124 | C22×C4 |
| Frattini, cosocle | C22×#124 | C23 |
| prime radical | C22×H(5,𝔽2) | D4 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C22×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1668 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)(3,9)(4,20,25,57)(5,28)(6,30)(7,33)(8,14)(10,23,42,59)(11,29)(12,32)(13,22)(15,49,51,64)(16,36)(17,45)(18,38)(19,39)(21,58,55,62)(24,43,31,44)(26,41,27,47)(34,48)(35,50,61,60)(37,56)(40,52,63,53)(46,54);(1,3)(2,9)(4,21,26,50)(5,17,29,56)(6,16)(7,34,22,18)(8,19)(10,40,43,64)(11,45,28,37)(12,46)(13,48,33,38)(14,39)(15,24,52,59)(20,35,41,62)(23,51,31,53)(25,55,27,60)(30,36)(32,54)(42,63,44,49)(47,58,57,61);(1,4,22,26,8,25,7,27)(2,10,13,44,14,42,33,43)(3,15,34,52,19,51,18,53)(5,23,32,24,29,59,6,31)(9,35,38,58,39,61,48,62)(11,20,30,41,28,57,12,47)(16,50,56,21,54,60,17,55)(36,49,45,40,46,64,37,63)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=defi,d^2=ij,e^2=i,h^2=j,b^a=bfg,c^a=cd,c^b=ce,d^a=di,d^b=dh,e^a=ej,e^b=ehij,e^c=ej,f^c=fj,g^c=gh,g^d=gi,g^e=gij,h^a=hi,h^b=hij,h^g=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72286330, #72286456, #72286649 |
| maximal normal subgroup of | #72286330, #72286456, #72286649 |
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.6);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.4*F.5*F.6*F.9);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
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);