Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#2127 |
| commutator | C42 | C24×C4 |
| Frattini, cosocle | #127 | C25 |
| socle | C2 | C22×#2127 |
| prime radical | #10575515 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C24×C4 ◁ C2×#3449 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
5643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,8,18,31)(4,20,23,9)(5,25,26,10)(6,11,14,12)(7,13)(15,28,41,53)(16,29,43,54)(17,30,19,32)(21,46,44,56)(22,33)(24,34)(27,36)(35,39)(37,38)(40,52,60,63)(42,51,45,55)(47,49)(48,62,61,64)(50,58,59,57);(1,3,14,19)(2,8,13,30)(4,21,24,41)(5,16,39,43)(6,18,7,17)(9,28)(10,29,25,51)(11,31)(12,32)(15,38,44,23)(20,46)(22,48,50,61)(26,42,27,45)(33,52)(34,53,37,56)(35,55)(36,54)(40,59,60,49)(47,62)(57,64,58,63);(1,4)(2,9)(3,15)(5,22)(6,24)(7,23)(8,28)(10,33)(11,37)(12,34)(13,20)(14,38)(16,40)(17,21)(18,44)(19,41)(25,47)(26,49)(27,50)(29,52)(30,46)(31,56)(32,53)(35,58)(36,57)(39,59)(42,48)(43,61)(45,60)(51,62)(54,64)(55,63);(1,5)(2,10)(3,16)(4,22)(6,27)(7,26)(8,29)(9,33)(11,35)(12,36)(13,25)(14,39)(15,40)(17,42)(18,45)(19,43)(20,47)(21,48)(23,49)(24,50)(28,52)(30,51)(31,55)(32,54)(34,57)(37,58)(38,59)(41,61)(44,60)(46,62)(53,64)(56,63);(3,17)(8,30)(15,21)(16,42)(18,19)(28,46)(29,51)(31,32)(40,48)(41,44)(43,45)(52,62)(53,56)(54,55)(60,61)(63,64)Polycyclic presentation:
2^10|a^2=f,b^2=ghj,h^2=j,b^a=bfg,c^a=ci,c^b=ci,d^a=di,e^b=ej,f^b=fh,f^c=fj,f^d=fj,g^a=ghij,g^b=gi,g^c=gj,g^f=gi,h^a=hi,h^b=hi,h^f=hj,h^g=hj,i^a=ij,i^b=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72284492, #72286910, #72287241, … |
| maximal normal subgroup of | #72284492, #72286910, #72287241, … |
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.8*F.10);;
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.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, 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, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, 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);