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 | #32898 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C22×#41 | C24 |
| socle | C2 | #392603 |
| prime radical | #62811 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
5058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(13,36)(17,43)(25,37)(29,55)(40,50)(44,62)(52,61)(56,64);(1,2,5,10,7,12,23,34)(3,8,14,26,16,28,42,54)(4,9,19,30,21,32,46,57)(6,24,22,47,18,11,35,33)(13,37,40,61,36,25,50,52)(15,27,41,53,39,48,58,63)(17,29,44,56,43,55,62,64)(20,31,38,51,45,49,59,60);(1,3,5,14,7,16,23,42)(2,8,10,26,12,28,34,54)(4,17,19,44,21,43,46,62)(6,15,22,41,18,39,35,58)(9,29,30,56,32,55,57,64)(11,27,33,53,24,48,47,63)(13,38,40,45,36,59,50,20)(25,51,52,49,37,60,61,31);(1,4,18,45)(2,9,24,49)(3,13,39,43)(5,19,35,59)(6,20,7,21)(8,25,48,55)(10,30,47,60)(11,31,12,32)(14,40,58,62)(15,17,16,36)(22,38,23,46)(26,52,63,64)(27,29,28,37)(33,51,34,57)(41,44,42,50)(53,56,54,61)Polycyclic presentation:
2^10|b^2=e,c^2=e,d^2=fj,e^2=j,c^a=cg,c^b=ch,d^a=dgij,d^b=dh,d^c=df,f^b=fi,g^d=gj,h^d=hi,h^f=hj,i^c=ij,i^d=ij106 conjugacy classes — view the full character table.
| maximal subgroup of | #72317393, #72317506, #72317598, … |
| maximal normal subgroup of | #72317393, #72317506, #72317598, … |
| maximal quotient of | C2×#72317366 |
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.5);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);