Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #23900 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C24 |
| prime radical | #6499871 | C2 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
3806 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 3806 conjugacy classes of subgroups — too many to draw. Only its 163 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)(3,12)(4,11)(5,21)(6,10)(7,13)(8,14)(9,16)(15,28)(17,25)(18,30)(19,26)(20,29)(22,24)(23,31)(27,32);(1,3,6,16)(2,9,10,12)(4,18,23,32)(5,8,17,7)(11,27,31,30)(13,25,14,21)(15,19,22,20)(24,26,28,29);(2,10)(4,15,23,22)(9,12)(11,28,31,24)(13,14)(18,19,32,20)(21,25)(26,27,29,30);(1,4)(2,11)(3,15)(5,22)(6,19)(7,23)(8,20)(9,24)(10,26)(12,30)(13,31)(14,29)(16,32)(17,18)(21,27)(25,28)Polycyclic presentation:
2^10|b^2=hj,c^2=fhj,e^2=ij,b^a=bhj,c^a=cf,d^a=dg,d^b=dg,d^c=de,e^a=eh,e^b=ehi,e^d=eij,f^d=fi,g^b=gj,g^c=ghij,g^e=gj,h^d=hj64 conjugacy classes — view the full character table.
| maximal subgroup of | #72234752, #72234789, #72237818, … |
| maximal normal subgroup of | #72234752, #72234789, #72237818, … |
| maximal quotient of | #72284479, #72285910, #72338709, … |
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, 2, F.8*F.10);;
gap> SetPower(coll, 3, F.6*F.8*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);