Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139700 |
| commutator | C2×#127 | C22×C4 |
| Frattini, cosocle | C2×#535 | C23 |
| prime radical | #499831 | C2 |
| LCS | G ▷ C2×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1869 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 1869 conjugacy classes of subgroups — too many to draw. Only its 73 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)(3,7)(4,15,16,25)(5,9)(6,10)(8,17,23,18)(11,26,28,32)(12,29)(13,21)(14,22)(19,24)(20,27,31,30);(1,3)(2,7)(4,11)(5,13)(6,14)(8,20)(9,21)(10,22)(12,19)(15,26)(16,28)(17,27)(18,30)(23,31)(24,29)(25,32);(3,11)(7,20)(9,24)(12,27)(13,30)(14,28)(15,25)(21,26)(22,31)(29,32)Polycyclic presentation:
2^10|a^2=f,g^2=j,c^a=ce,c^b=cd,d^a=dg,e^a=eh,e^b=egij,e^d=eij,f^c=fh,f^d=fj,g^c=gi,g^d=gj,g^e=gj,h^b=hj,i^a=ij,i^b=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72241311, #72251744, #72251760, … |
| maximal normal subgroup of | #72241311, #72251744, #72251760, … |
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, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*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);