Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #10336 |
| commutator, Frattini, cosocle | C22×#120 | C23 |
| prime radical | #245225 | C2 |
| LCS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
2656 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 2656 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(1,2,8,24)(3,14,6,13)(4,18,19,27)(5,11,7,12)(9,17,29,15)(10,16,30,21)(20,32,23,26)(22,25,31,28);(2,9,12,14)(3,15)(4,19,20,23)(5,8)(10,18)(11,29,24,13)(16,31,22,21)(25,27)(26,30)(28,32);(2,10,12,28)(3,8)(4,20)(5,15)(9,25,14,30)(11,26,24,27)(13,18,29,32)(16,23)(19,22)(21,31)Polycyclic presentation:
2^10|a^2=gj,b^2=egij,c^2=h,b^a=be,c^a=cf,c^b=cd,d^a=dg,d^b=dh,e^a=ej,e^b=ej,e^c=ehi,f^a=fi,f^b=fghj,f^c=fj,f^d=fi,f^e=fj,g^b=gj,g^c=gi,h^a=hj52 conjugacy classes — view the full character table.
| maximal subgroup of | #72227793, #72236915, #72262416, … |
| maximal normal subgroup of | #72227793, #72236915, #72262416, … |
| maximal quotient of | #72237580, #72237605, #72262495, … |
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*F.10);;
gap> SetPower(coll, 2, F.5*F.7*F.9*F.10);;
gap> SetPower(coll, 3, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);