Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #115995 |
| 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 | |
1890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,7,12,25,40)(3,13,15,8,18,31,16,33)(4,19,21,38,23,9,34,36)(6,26,28,39,29,11,24,42)(14,43,45,30,47,60,46,61)(17,44,49,41,27,59,48,32)(20,52,53,50,55,62,54,35)(22,37,57,64,58,51,56,63);(1,3)(2,8,12,33)(4,20)(5,15,25,16)(6,27,29,17)(7,18)(9,35,19,50)(10,13)(11,41)(14,22,47,58)(21,53,34,54)(23,55)(24,49)(26,32)(28,48)(30,51)(31,40)(36,62)(37,61)(38,52)(39,59,42,44)(43,64,60,63)(45,56)(46,57);(1,4,7,23)(2,9,12,19)(3,14,18,47)(5,21,25,34)(6,22,29,58)(8,30,33,61)(10,38,40,36)(11,37,26,51)(13,43,31,60)(15,46,16,45)(17,20,27,55)(24,56,28,57)(32,35,41,50)(39,64,42,63)(44,52,59,62)(48,53,49,54)Polycyclic presentation:
2^10|a^2=d,b^2=g,c^2=j,d^2=j,b^a=be,c^a=cdehj,c^b=cf,d^b=dh,d^c=dhj,e^a=eh,e^b=ehj,e^c=ehj,e^d=ej,f^a=fhj,f^b=fhi,f^d=fj,f^e=fj,g^a=ghj,g^c=ghi,g^d=gj,g^e=gj,g^f=gj,h^a=hj,h^b=hj,i^a=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72311163, #72311253, #72311401, … |
| maximal normal subgroup of | #72311163, #72311253, #72311401, … |
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.7);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.5*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*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.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);