Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #32898 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C24 |
| prime radical | C27 | D4 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
7747 conjugacy classes of subgroups; only the 241 normal subgroups are shown.
This group has 7747 conjugacy classes of subgroups — too many to draw. Only its 241 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 — 4 generators on 32 points:
(1,2,8,13)(3,15,21,32)(4,11,6,25)(5,23,18,28)(7,26,22,30)(9,14,16,27)(10,24,29,20)(12,17,31,19);(1,3,16,5)(2,10,27,31)(4,20,7,17)(6,24,22,19)(8,21,9,18)(11,23,26,15)(12,13,29,14)(25,28,30,32);(1,4,16,7)(2,11,27,26)(3,17,5,20)(6,9,22,8)(10,28,31,32)(12,15,29,23)(13,25,14,30)(18,24,21,19);(1,5,16,3)(2,12,27,29)(4,17,7,20)(6,19,22,24)(8,18,9,21)(10,13,31,14)(11,23,26,15)(25,28,30,32)Polycyclic presentation:
2^10|a^2=h,b^2=hj,c^2=hj,d^2=hj,f^2=h,g^2=hj,b^a=bg,c^a=ce,c^b=chi,d^a=defh,d^c=dhj,e^b=ej,e^d=ej,f^a=fh,f^b=fhi,f^c=fj,f^d=fhj,g^a=ghj,g^b=ghj,g^d=ghi,g^f=gj,i^a=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72227603, #72227899 |
| maximal normal subgroup of | #72227899 |
| maximal quotient of | #72227598, #72227689 |
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.8);;
gap> SetPower(coll, 2, F.8*F.10);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);