Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C4×C8 | C25 |
| prime radical | #7623529 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ #17316 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,10)(5,20)(6,9)(7,11)(8,12)(14,27)(15,23)(16,28)(17,25)(18,22)(19,26)(21,24)(29,31)(30,32)(33,34)(35,45)(36,42)(37,43)(38,44)(39,40)(41,47)(46,57)(48,58)(49,61)(50,64)(51,55)(52,54)(53,56)(59,62)(60,63);(1,3,7,16,8,6,5,15)(2,9,11,23,12,13,20,28)(4,14,17,29,19,18,21,30)(10,22,25,32,26,27,24,31)(34,40)(35,46,47,48)(36,37)(38,53)(41,57,45,58)(42,55)(43,54)(49,63,62,50)(51,52)(59,60,61,64);(1,4)(2,10)(3,14)(5,21)(6,18)(7,17)(8,19)(9,22)(11,25)(12,26)(13,27)(15,30)(16,29)(20,24)(23,32)(28,31)(34,40)(35,47)(42,54)(43,55)(44,56)(46,48)(49,62)(50,63);(1,5,8,7)(2,11,12,20)(3,15,6,16)(4,17,19,21)(9,23,13,28)(10,24,26,25)(14,29,18,30)(22,31,27,32)(33,35,37,46,38,49,36,50,39,47,51,48,53,62,52,63)(34,41,43,58,44,59,42,60,40,45,55,57,56,61,54,64);(1,6,7,15,8,3,5,16)(2,9,11,23,12,13,20,28)(4,18,17,30,19,14,21,29)(10,22,25,32,26,27,24,31)(33,36,38,51,39,52,53,37)(34,42,44,55,40,54,56,43)(35,48,62,50,47,46,49,63)(41,57,61,60,45,58,59,64)Polycyclic presentation:
2^10|b^2=f,d^2=gh,e^2=fi,f^2=h,g^2=i,i^2=j,b^a=bhj,c^a=cj,d^a=dhj,d^b=dg,d^c=dhj,e^b=eij,e^d=eij,f^d=fj,g^b=gi,g^e=gj,i^b=ij106 conjugacy classes — view the full character table.
| maximal subgroup of | #72287479, #72287605, #72287659, … |
| maximal normal subgroup of | #72287479, #72287605, #72287659, … |
| maximal quotient of | #72343464, #72343487, #72343507, … |
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.6);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 5, F.6*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);