Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #213828 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | #6499874 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
2985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,16,19,27)(5,20)(6,7)(8,14)(9,25,23,18)(10,21)(12,22)(13,30)(15,28)(17,29)(24,31)(26,32);(2,7)(3,12)(8,11)(9,23)(10,20)(13,17)(14,15)(16,27)(22,28)(24,29)(26,30)(31,32);(1,3,5,12)(2,8,10,28)(4,17,18,13)(6,15,21,14)(7,22,20,11)(9,26,27,24)(16,30,23,29)(19,32,25,31);(1,4,6,19)(2,9,7,23)(3,13,15,31)(5,18,21,25)(8,24,22,30)(10,27,20,16)(11,29,28,26)(12,17,14,32)Polycyclic presentation:
2^10|a^2=g,c^2=ef,d^2=j,e^2=j,b^a=bj,c^a=ce,c^b=cef,d^a=defij,d^c=def,e^a=eh,e^b=ei,e^c=ei,e^d=ei,f^a=fhij,f^b=fi,f^c=fi,f^d=fi,f^e=fj,g^c=ghj,g^d=gij,h^d=hj,i^c=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72258243, #72262472, #72262571, … |
| maximal normal subgroup of | #72258243, #72262472, #72262571, … |
| maximal quotient of | #72258350, #72258387, #72338766, … |
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.7);;
gap> SetPower(coll, 3, F.5*F.6);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);