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 | #9495 |
| commutator | C25 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| socle | C2 | C2×#9495 |
| prime radical | #6437616 | C2 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
4812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(4,15)(8,20)(10,17)(11,24)(12,26)(16,18)(27,28);(1,2,6,10)(3,11,14,8)(4,9,16,23)(5,17)(7,19)(12,25,27,22)(13,20)(15,21)(18,30)(24,29)(26,32)(28,31);(1,3,6,14)(2,8,10,11)(4,12,16,27)(5,13,19,29)(7,20,17,24)(9,22,23,25)(15,26,18,28)(21,31,30,32);(1,4,6,16)(2,9)(3,12,14,27)(5,18,19,15)(7,21)(8,22)(10,23)(11,25)(13,28,29,26)(17,30)(20,31)(24,32)Polycyclic presentation:
2^10|b^2=g,c^2=j,d^2=i,b^a=be,c^b=cj,d^a=def,d^b=dhi,e^b=eh,e^d=ei,f^b=fhi,f^d=fi,g^a=gh,g^e=gj,h^b=hj,h^d=hj,i^b=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72233466, #72258313, #72258400, … |
| maximal normal subgroup of | #72233466, #72258313, #72258400, … |
| maximal quotient of | C4×#139750, C2×#72233461 |
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.7);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);