Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #93182 |
| commutator | C22×C22⋊C4 | C42 |
| Frattini, prime radical, cosocle | C2×#2613 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2613 ◁ G | |
| LECS | G ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
1242 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 1242 conjugacy classes of subgroups — too many to draw. Only its 41 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 — 2 generators on 32 points:
(1,2,5,9)(3,11,13,27,14,29,31,28)(4,15,16,25)(6,10,19,26)(7,20,21,30,23,32,22,12)(8,17,24,18);(1,3,5,13)(2,7,9,22)(4,12,17,20)(6,14,19,31)(8,11,25,28)(10,23,26,21)(15,29,24,27)(16,32,18,30)Polycyclic presentation:
2^10|a^2=d,b^2=e,c^2=hij,d^2=h,b^a=bc,c^a=cf,c^b=cg,d^b=dfhij,d^c=di,e^a=eghij,e^c=ehij,e^d=eh,f^a=fi,f^b=fh,f^c=fj,f^d=fj,f^e=fj,g^a=ghj,g^b=ghij,h^b=hj,i^a=ij,i^b=ij34 conjugacy classes — view the full character table.
| maximal subgroup of | #72238375, #72238381, #72238382, … |
| maximal normal subgroup of | #72238375, #72238381, #72238382, … |
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.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9*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> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);