Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #119556 |
| commutator | C22×#41 | C22×C4 |
| Frattini, cosocle | C22×H(5,𝔽2) | C23 |
| prime radical | #35527 | C22 |
| LCS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
1590 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 1590 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11)(3,14,10,15)(4,19,22,29)(5,17,26,18)(7,12,8,13)(9,21,27,31)(16,23,20,24)(25,30,32,28);(1,3,15,13)(2,8,18,26)(4,20,31,30)(5,11,6,10)(7,17,14,12)(9,22,16,25)(19,21,28,23)(24,29,32,27);(1,4,5,25,8,24,14,21)(2,9,17,20,13,28,10,29)(3,16,12,27,18,19,11,30)(6,22,15,31,7,23,26,32)Polycyclic presentation:
2^10|a^2=e,b^2=dehi,c^2=dij,d^2=j,g^2=j,b^a=bd,c^a=cdf,c^b=cg,d^a=dh,d^b=dhj,d^c=dj,e^b=ehj,e^c=ei,e^d=ej,f^a=fhij,f^b=fj,f^c=fhj,g^a=gh,g^b=gij,g^c=ghj,g^e=gj,g^f=gj,h^a=hj,h^b=hj,i^a=ij,i^b=ij,i^c=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72235241, #72285838, #72286251, … |
| maximal normal subgroup of | #72235241, #72285838, #72286251, … |
| maximal quotient of | #72235267 |
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.5);;
gap> SetPower(coll, 2, F.4*F.5*F.8*F.9);;
gap> SetPower(coll, 3, F.4*F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*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> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);