Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #119911 |
| commutator | C23×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×H(5,𝔽2) | C23 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
1861 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 1861 conjugacy classes of subgroups — too many to draw. Only its 135 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)(3,14)(4,19,24,39)(5,25)(6,12)(7,11)(8,13)(9,16)(10,22,37,27)(15,44,47,64)(17,43)(18,33)(20,54,45,61)(21,53,23,52)(26,40)(28,41)(29,34)(30,38)(31,55,60,57)(32,48)(35,49,42,46)(36,58,59,56)(50,62)(51,63);(1,3)(2,9)(4,20)(5,16,7,29)(6,17)(8,18)(10,35)(11,14,25,33)(12,32)(13,34)(15,23,47,21)(19,44)(22,57)(24,45)(26,48,30,50)(27,55)(28,51)(31,36,60,59)(37,42)(38,43,40,63)(39,64)(41,62)(46,56,49,58)(52,54,53,61);(1,4,8,24)(2,10)(3,15,18,47)(5,21,7,23)(6,27)(9,31)(11,36)(12,39,41,19)(13,37)(14,42)(16,20,29,45)(17,49)(22,28)(25,59)(26,58)(30,56)(32,61,62,54)(33,35)(34,60)(38,53,40,52)(43,64,63,44)(46,51)(48,57)(50,55)Polycyclic presentation:
2^10|a^2=h,b^2=g,c^2=dfgij,d^2=j,e^2=j,b^a=bd,c^a=ce,c^b=cf,d^a=dj,d^b=di,d^c=dij,e^b=eh,e^c=eh,f^a=fhij,f^b=fij,f^d=fj,f^e=fj,g^a=gij,g^c=gij,g^d=gj,g^f=gj,h^c=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72341307, #72341395, #72341450, … |
| maximal normal subgroup of | #72341307, #72341395, #72341450, … |
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.8);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.4*F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
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.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);