Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #10336 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C23×#41 | C23 |
| socle | C2 | #116351 |
| prime radical | #62811 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
2919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10)(3,13,15,37)(4,9,19,30)(6,24,22,51)(7,12,23,34)(8,25,27,53)(11,35,33,18)(14,39,40,62)(16,46,44,28)(17,38,45,29)(20,50,48,61)(21,32,49,59)(26,52,54,64)(31,60,58,47)(36,43,56,42)(41,57,63,55);(1,3)(2,8,12,28)(4,14)(5,15)(6,16)(7,17)(9,26,32,55)(10,27,34,46)(11,29,24,37)(13,33,38,51)(18,25)(19,40)(20,41)(21,42)(22,44)(23,45)(30,54,59,57)(31,56,50,62)(35,53)(36,61,39,58)(43,49)(47,52)(48,63)(60,64);(1,4,18,47,7,21,6,20)(2,9,11,31,12,32,24,50)(3,14,25,52,17,42,16,41)(5,19,35,60,23,49,22,48)(8,26,29,56,28,55,37,62)(10,30,33,58,34,59,51,61)(13,36,27,57,38,39,46,54)(15,43,53,63,45,40,44,64)Polycyclic presentation:
2^10|a^2=d,b^2=hi,c^2=fhij,f^2=j,b^a=bdfj,c^a=ce,c^b=cg,d^c=dh,e^a=eh,e^b=ehi,f^a=fj,g^a=gij,g^d=gj,h^b=hj,i^a=ij,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72317504, #72317506, #72317617, … |
| maximal normal subgroup of | #72317504, #72317506, #72317617, … |
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.4);;
gap> SetPower(coll, 2, F.8*F.9);;
gap> SetPower(coll, 3, F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.6*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, 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);