Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #8376 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C2×#131 | C24 |
| socle | C2 | C2×#8376 |
| prime radical | #10576609 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
3770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,9)(3,13)(4,18,20,10)(5,22,25,56)(6,26,28,12)(8,29)(11,36,33,24)(14,41)(15,43)(16,31)(17,47)(19,52,54,39)(21,35,48,51)(23,32)(27,37)(30,44)(34,46,50,58)(38,57)(40,42)(45,61)(49,59)(53,62)(55,63)(60,64);(1,3,4,15)(2,8,10,30)(5,23,19,53)(6,27,21,55)(7,14,20,42)(9,31,18,49)(11,37,34,61)(12,38,35,62)(13,39,40,22)(16,46,44,24)(17,48,45,28)(25,57,54,64)(26,32,51,60)(29,58,59,36)(33,47,50,63)(41,52,43,56);(2,9)(3,14)(10,18)(11,33)(12,26)(15,42)(16,29)(17,27)(22,56)(23,57)(34,50)(35,51)(39,52)(44,59)(45,55)(53,64);(3,15,14,42)(8,30,31,49)(13,40,41,43)(16,44,29,59)(17,45,27,55)(23,53,57,64)(32,60,38,62)(37,61,47,63)Polycyclic presentation:
2^10|a^2=hj,b^2=e,d^2=h,f^2=i,i^2=j,b^a=bf,c^a=cj,c^b=cj,d^a=dh,d^b=deg,e^a=ej,e^d=eh,f^a=fi,f^b=fi,g^b=ghj,g^d=gh,g^e=gj,h^b=hj,i^a=ij,i^b=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72307386, #72307633 |
| maximal normal subgroup of | #72307386, #72307633 |
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.8*F.10);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);