Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #496225 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C2×#127 | C24 |
| prime radical | #6437457 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ C2×#3500 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,8)(4,18,20,9)(5,22,24,36,25,38,26,10)(6,12)(7,11)(13,27)(14,39,41,28)(15,43,45,29,44,35,23,54)(16,30)(17,37)(19,48,50,47,52,55,51,32)(21,34,53,33)(31,46)(40,58,49,56,61,64,62,60)(42,57,63,59);(1,3)(2,8)(4,14)(5,23,25,45)(6,17)(7,16)(9,28)(10,35,36,43)(11,30)(12,37)(13,31)(15,26,44,24)(18,39)(19,49,52,62)(20,41)(21,42)(22,54,38,29)(27,46)(32,58,47,64)(33,59)(34,57)(40,50,61,51)(48,56,55,60)(53,63);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,28)(10,32)(11,18)(12,33)(13,34)(15,40)(16,41)(17,42)(22,55)(23,49)(24,51)(25,52)(26,50)(27,53)(29,56)(30,39)(31,57)(35,58)(36,47)(37,59)(38,48)(43,64)(44,61)(45,62)(46,63)(54,60);(1,5,7,25)(2,10,11,36)(3,15,16,44)(4,19,20,52)(6,26,27,24)(8,29,30,54)(9,32,18,47)(12,22,13,38)(14,40,41,61)(17,23,46,45)(21,50,53,51)(28,56,39,60)(31,43,37,35)(33,55,34,48)(42,49,63,62)(57,64,59,58)Polycyclic presentation:
2^10|a^2=g,b^2=ij,d^2=j,g^2=ij,h^2=j,c^a=ce,d^a=df,d^b=dehj,e^a=ej,f^a=fh,f^b=fi,g^c=gj,g^d=ghj,g^f=gij,h^a=hi,h^d=hj,h^f=hj,i^d=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72306893, #72307002, #72317996, … |
| maximal normal subgroup of | #72306893, #72307002, #72317996, … |
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.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);