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, socle | C2 | #148463 |
| commutator | C2×#124 | C22×C4 |
| Frattini, cosocle | C22×#124 | C23 |
| prime radical | C22×H(5,𝔽2) | D4 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C22×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1618 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)(4,18,22,48)(5,10)(6,26)(7,13)(9,21,36,56)(11,28)(12,25)(14,41,45,64)(15,32)(16,31)(17,33)(19,50,53,61)(20,47,55,49)(23,30)(24,40)(27,39)(29,44,60,51)(34,52,63,43)(35,54,37,42)(38,58)(46,62)(57,59);(1,3)(2,8)(4,19,20,44)(5,23,25,15)(6,27,28,57)(7,17)(9,34,37,41)(10,32,12,30)(11,39,26,59)(13,33)(14,42,43,21)(16,24)(18,29,47,61)(22,53,55,51)(31,40)(35,64,36,63)(38,62)(45,54,52,56)(46,58)(48,60,49,50);(1,4,5,20,7,22,25,55)(2,9,10,35,13,36,12,37)(3,14,15,43,17,45,23,52)(6,21,24,54,28,56,58,42)(8,29,30,50,33,60,32,61)(11,18,38,47,26,48,40,49)(16,44,27,19,46,51,57,53)(31,41,59,63,62,64,39,34)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=d,d^2=j,e^2=ij,h^2=j,b^a=bfg,c^a=ce,c^b=cefhij,d^a=dij,d^b=dij,e^a=ei,e^b=eh,f^c=fj,g^c=ghj,g^d=gj,g^e=gi,h^a=hi,h^b=hij,h^g=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72286330, #72286528, #72286616 |
| maximal normal subgroup of | #72286330, #72286528, #72286616 |
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.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*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*F.10);;
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);