Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8101 |
| commutator | C23×C4 | C2×C42 |
| Frattini, cosocle | C22×#120 | C23 |
| prime radical | C22×H(3,𝔽4) | C22 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×H(3,𝔽4) ◁ G | |
| LECS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
2016 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 2016 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,29,12,7,32)(3,16,18,39,19,37,20,40)(4,10,24,42,59,44,60,48)(6,14,30,47,8,15,31,13)(9,21,36,28,38,22,50,33)(17,51,52,64,23,57,58,63)(25,45,34,61,27,43,26,46)(35,55,62,53,41,54,49,56);(1,3,5,28)(2,9,32,50)(4,23,26,52)(6,22,30,20)(7,33,29,19)(8,21,31,18)(10,41,61,51)(11,36,12,38)(13,16,14,39)(15,40,47,37)(17,34,58,59)(24,53,27,54)(25,55,60,56)(35,46,57,44)(42,49,45,63)(43,64,48,62);(1,4)(2,10)(3,17)(5,26)(6,27)(7,34)(8,25)(9,35)(11,46)(12,44)(13,48)(14,43)(15,45)(16,49)(18,53)(19,23)(20,56)(21,54)(22,55)(24,30)(28,58)(29,59)(31,60)(32,61)(33,52)(36,51)(37,62)(38,41)(39,63)(40,64)(42,47)(50,57)Polycyclic presentation:
2^10|a^2=d,b^2=efj,d^2=ij,f^2=j,b^a=bde,c^a=cgi,c^b=cij,d^b=dh,d^c=di,e^a=ehi,e^b=eg,e^c=eij,e^d=ej,f^a=fg,f^b=fg,f^c=fij,f^d=fi,f^e=fj,g^a=gi,g^b=gj,h^a=hj,h^b=hij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72235372, #72235395 |
| maximal normal subgroup of | #72235372, #72235395 |
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.4);;
gap> SetPower(coll, 2, F.5*F.6*F.10);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);