Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #392667 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#32948 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ C2×#3573 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
5454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8)(5,22)(6,23)(9,25)(10,26)(11,13)(12,36)(14,37)(15,41)(17,42)(18,34)(19,45)(21,46)(24,47)(27,28)(29,52)(30,31)(32,55)(33,56)(35,57)(38,48)(39,60)(43,62)(44,51)(49,50)(53,61)(54,63)(59,64);(1,2,5,11,23,27,47,28,7,13,24,8,6,12,22,36)(3,9,15,30,42,49,62,50,16,31,43,25,17,32,41,55)(4,18,19,26,46,35,56,57,20,10,33,34,21,44,45,51)(14,38,39,48,58,54,60,64,37,29,53,52,40,59,61,63);(1,3,7,16)(2,9,13,31)(4,14,20,37)(5,15,24,43)(6,17,23,42)(8,25,11,30)(10,29,18,38)(12,32,27,49)(19,39,33,53)(21,40,46,58)(22,41,47,62)(26,48,34,52)(28,50,36,55)(35,54,44,59)(45,61,56,60)(51,63,57,64);(1,4,7,20)(2,10,13,18)(3,14,16,37)(5,19,24,33)(6,21,23,46)(8,26,11,34)(9,29,31,38)(12,35,27,44)(15,39,43,53)(17,40,42,58)(22,45,47,56)(25,48,30,52)(28,51,36,57)(32,54,49,59)(41,61,62,60)(50,63,55,64)Polycyclic presentation:
2^10|b^2=e,c^2=j,d^2=j,e^2=ij,f^2=ij,i^2=j,b^a=bf,c^a=ch,d^a=dg,d^b=dj,e^a=ei,f^a=fi,f^b=fj,g^c=gj,h^d=hj,i^a=ij154 conjugacy classes — view the full character table.
| Fitting subgroup of | #72341844 |
| maximal subgroup of | #72320817, #72341844, #72341903, … |
| maximal normal subgroup of | #72320817, #72341844, #72343925, … |
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, 2, F.5);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);