Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×#442 |
| commutator | C22×C8 | C24×C4 |
| Frattini, cosocle | C2×C4×C8 | C25 |
| prime radical | #72225017 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ C2×#17320 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
10978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(21,22,23,26,24,27,29,34,25,28,30,33,31,35,36,32);(5,6,9,12,10,13,14,7)(8,11,17,19,16,18,20,15)(22,26,28,33)(23,29)(24,31)(27,32,35,34)(30,36);(5,7,14,13,10,12,9,6)(8,15,20,18,16,19,17,11)(22,27,28,35)(23,30)(26,32,33,34)(29,36);(5,8)(6,11)(7,15)(9,17)(10,16)(12,19)(13,18)(14,20);(2,4)(8,16)(11,18)(15,19)(17,20)Polycyclic presentation:
2^11|a^2=h,b^2=f,c^2=fi,f^2=i,h^2=j,j^2=k,b^a=bhjk,c^a=cjk,e^a=eg,e^d=ei,f^a=fk,h^b=hj,h^c=hk,j^b=jk182 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72225133, #72319695, #72322017, … |
| maximal normal subgroup of | C2×#72225133, #72319695, #72322017, … |
| maximal quotient of | #72225134, #72225136, C2×#72225133, … |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);