Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #106766 |
| commutator | C22×D4 | C2×C42 |
| Frattini, cosocle | C22×H(5,𝔽2) | C23 |
| prime radical | C2×#3500 | C22 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
2995 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)(5,10)(6,27,28,11)(7,12)(9,21,36,56)(13,41)(14,30)(15,32,16,31)(17,33)(19,49)(20,53)(22,51)(23,50)(24,39,25,38)(26,40)(29,42,58,43)(34,52,59,64)(35,54,37,55)(44,57)(45,63)(46,62,48,60)(47,61);(1,3)(2,8)(4,19,22,45)(5,14)(6,16)(7,17)(9,34,36,59)(10,30)(11,32)(12,33)(13,20,44,23)(15,28)(18,49,51,63)(21,52,56,64)(24,48)(25,46)(26,47)(27,31)(29,35,58,37)(38,62)(39,60)(40,61)(41,53,57,50)(42,54,43,55);(1,4)(2,9)(3,13,17,44)(5,23,26,20)(6,21)(7,22)(8,29,33,58)(10,37,40,35)(11,18)(12,36)(14,45)(15,42,16,43)(19,47)(24,55,25,54)(27,51)(28,56)(30,59)(31,41,32,57)(34,61)(38,50,39,53)(46,64)(48,52)(49,62)(60,63)Polycyclic presentation:
2^10|a^2=g,b^2=h,c^2=ef,d^2=j,c^a=ce,c^b=cd,d^a=dh,d^c=dh,e^a=ei,e^b=ehj,e^d=ej,f^a=fi,g^c=gi,g^e=gj,g^f=gj,h^c=hj,i^a=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72302986, #72303101, #72341307, … |
| maximal normal subgroup of | #72302986, #72303101, #72341307, … |
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.7);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.5*F.6);;
gap> SetPower(coll, 4, F.10);;
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.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, 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, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);