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, socle | C22 | #10965 |
| commutator | C22×C42 | C22×C4 |
| Frattini, cosocle | C22×C4×D4 | C23 |
| prime radical | #4421184 | C2 |
| LCS | G ▷ C22×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#546 ◁ G | |
| LECS | G ▷ C22×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
3722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,13,8,3,6,12)(4,17,16,19,30,26,11,29)(5,20,24,21,25,23,18,22)(9,15,28,31,10,14,27,32);(1,3,8,2)(4,18,22,29)(5,23,17,11)(6,12,7,13)(9,14,10,15)(16,25,20,26)(19,30,24,21)(27,32,28,31);(1,4,19,15)(2,9,22,24)(3,10,21,18)(5,13,7,11)(6,16,25,12)(8,30,29,14)(17,31,28,23)(20,26,32,27)Polycyclic presentation:
2^10|a^2=f,b^2=efij,c^2=dfi,d^2=i,e^2=ij,f^2=ij,b^a=be,c^a=cd,c^b=cdegij,d^a=dh,d^b=dh,d^c=dhi,e^b=eij,e^c=ei,f^b=fij,f^c=fhi,f^d=fj,g^a=gi,g^b=gi,g^c=gi,h^a=hj,h^b=hj,h^c=hj64 conjugacy classes — view the full character table.
| maximal subgroup of | #72234079, #72234090, #72234092, … |
| maximal normal subgroup of | #72234079, #72234090, #72234092, … |
| maximal quotient of | C2×#72234076 |
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.6);;
gap> SetPower(coll, 2, F.5*F.6*F.9*F.10);;
gap> SetPower(coll, 3, F.4*F.6*F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);