Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #151462 |
| commutator, Frattini, cosocle | #1781 | C23 |
| prime radical | #10336 | C22 |
| LCS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
524 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 37 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:
(3,11)(4,15)(5,18)(7,19)(8,16)(12,23)(13,24)(14,25)(17,29)(20,30)(26,32)(27,28);(1,2)(3,12,14,26)(4,16,15,8)(5,9)(6,10)(7,13,20,28)(11,23)(17,31,29,21)(18,22)(19,24)(25,32)(27,30);(1,3,6,14)(2,7,10,20)(4,11)(5,13)(8,19)(9,12)(15,25)(16,30)(17,24,29,27)(18,28)(21,32,31,23)(22,26)Polycyclic presentation:
2^10|b^2=hi,c^2=gh,d^2=g,e^2=j,b^a=bgij,c^a=cd,c^b=ce,d^a=dg,d^b=df,d^c=dgi,e^a=efhij,e^c=eij,e^d=ehij,f^a=fi,f^b=fj,f^c=fh,f^d=fi,g^b=gi,g^e=gj,h^a=hi,h^d=hj,i^c=ij31 conjugacy classes — view the full character table.
| maximal subgroup of | #72285713, #72286199, #72338654, … |
| maximal normal subgroup of | #72285713, #72286199, #72338654, … |
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.9);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);