Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #12763 |
| commutator | #532 | C22×C4 |
| Frattini, cosocle | C2×#532 | C23 |
| socle | C2 | #147826 |
| prime radical | C2×#3018 | C22 |
| 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 | |
1804 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)(4,18,21,9)(5,23)(6,12)(7,11)(10,25)(13,39,42,61)(14,44)(15,30)(16,38)(17,31)(19,51,54,62)(20,33,40,49)(22,50,52,34)(24,35)(26,59)(27,37)(28,41,60,43)(29,47)(32,55,48,53)(36,45)(46,58)(56,63,57,64);(1,3,7,17)(2,8,11,31)(4,19,52,55)(5,24)(6,15)(9,32,50,62)(10,35)(12,38)(13,40,43,56)(14,45,47,26)(16,27)(18,48,34,51)(20,41,57,42)(21,54,22,53)(23,58)(25,46)(28,49,39,63)(29,59,44,36)(30,37)(33,61,64,60);(1,4,6,22,7,21,27,52)(2,9,12,34,11,18,37,50)(3,13,16,43,17,42,15,41)(5,20,26,57,25,40,45,56)(8,28,30,39,31,60,38,61)(10,33,36,64,23,49,59,63)(14,19,24,55,47,54,46,53)(29,32,35,62,44,48,58,51)Polycyclic presentation:
2^10|a^2=f,b^2=gj,c^2=h,d^2=i,e^2=i,h^2=j,b^a=bfg,c^a=ce,c^b=cd,d^a=dj,d^b=dh,e^a=eij,e^b=ehij,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, #72286647, #72286971 |
| maximal normal subgroup of | #72284990, #72286647, #72286971 |
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*F.10);;
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.6*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*F.10);;
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);