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 | C4 | #12965 |
| commutator | #532 | C22×C4 |
| Frattini, prime radical, cosocle | C2×#532 | C23 |
| socle | C2 | #148238 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1216 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)(3,8)(4,17,20,9)(5,22)(6,12)(7,11)(10,24)(13,39,41,61)(14,43)(15,30)(16,32)(18,48,50,62)(19,34,52,46)(21,47,55,35)(23,36)(25,57)(26,31)(27,38)(28,40,60,42)(29,44)(33,51,45,49)(37,59)(53,63,54,64)(56,58);(1,3)(2,8)(4,18,21,51)(5,23,24,58)(6,26,27,15)(7,16)(9,33,35,62)(10,36,22,56)(11,32)(12,30,38,31)(13,19,42,53)(14,25)(17,45,47,48)(20,50,55,49)(28,34,39,63)(29,37)(40,54,41,52)(43,57)(44,59)(46,61,64,60);(1,4,6,21,7,20,27,55)(2,9,12,35,11,17,38,47)(3,13,15,40,16,41,26,42)(5,19,25,54,24,52,59,53)(8,28,31,61,32,60,30,39)(10,34,37,64,22,46,57,63)(14,18,23,49,44,50,58,51)(29,33,36,48,43,45,56,62)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,h^2=j,b^a=bg,c^a=ce,c^b=cd,d^a=dj,d^b=dh,e^a=eij,e^b=ehi,f^c=fj,g^c=ghij,g^d=gij,g^e=gi,h^a=hi,h^b=hi,h^g=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72284990, #72286526, #72287090 |
| maximal normal subgroup of | #72284990, #72286526, #72287090 |
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.8);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
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);