Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #110180 |
| commutator | C23×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×H(5,𝔽2) | C23 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
1531 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 1531 conjugacy classes of subgroups — too many to draw. Only its 151 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:
(1,2,5,10,7,12,24,37)(3,13,15,41,17,42,22,52)(4,9,19,32,21,34,50,59)(6,25,23,53,27,11,38,36)(8,28,30,16,31,46,35,26)(14,43,44,64,18,40,48,63)(20,33,49,58,51,39,62,61)(29,55,56,45,57,54,60,47);(1,3)(2,8)(4,18)(5,22,24,15)(6,26,27,16)(7,17)(9,29,34,57)(10,35,37,30)(11,13,25,42)(12,31)(14,21)(19,48,50,44)(20,47,51,45)(23,46)(28,38)(32,56)(33,43)(36,52)(39,40)(41,53)(49,55)(54,62)(58,64,61,63)(59,60);(1,4,7,21)(2,9)(3,14)(5,19,24,50)(6,20)(8,29)(10,32)(11,39,25,33)(12,34)(13,40,42,43)(15,44)(16,47,26,45)(17,18)(22,48)(23,49)(27,51)(28,54,46,55)(30,56)(31,57)(35,60)(36,61,53,58)(37,59)(38,62)(41,63,52,64)Polycyclic presentation:
2^10|a^2=d,b^2=f,c^2=ghj,d^2=j,e^2=j,b^a=be,c^a=chj,c^b=cfg,d^b=dh,e^a=ehj,e^b=ehi,e^c=ehij,f^a=fhij,f^c=fi,f^d=fj,f^e=fj,g^a=gj,g^b=gi,g^c=gj,g^e=gj,h^b=hj,h^c=hj,i^a=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72283861, #72284092, #72285090, … |
| maximal normal subgroup of | #72283861, #72284092, #72285090, … |
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.6);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);