Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #18178 |
| commutator, Frattini, cosocle | C2×#97 | C24 |
| prime radical | #7614034 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
2039 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 2039 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,6,8,7)(10,15,18,28)(12,24)(13,26)(16,30)(17,27,33,32)(19,36,35,29)(20,37,34,40)(22,39)(23,38)(25,31);(1,3,5,8)(2,6,4,7)(9,10,16,28,13,19,30,36,14,18,12,15,26,35,24,29)(11,20,25,40,23,17,31,27,21,34,22,37,38,33,39,32);(1,2,5,4)(3,6,8,7)(9,11)(10,17,18,33)(12,25)(13,23)(14,21)(15,27,28,32)(16,22)(19,34,35,20)(24,31)(26,38)(29,37,36,40)(30,39);(2,4)(3,7)(6,8)(10,18)(19,35)(27,32)(37,40)Polycyclic presentation:
2^10|a^2=gh,b^2=fhj,c^2=gh,e^2=h,f^2=i,i^2=j,b^a=bf,c^b=cij,d^a=dgh,d^b=de,d^c=dgh,e^a=eg,e^b=egh,e^c=ej,e^d=eh,f^a=fi,f^c=fj,f^d=fg,f^e=fj,g^b=gj,i^a=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72307513, #72307699, #72307751 |
| maximal normal subgroup of | #72307513, #72307699, #72307751 |
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.8);;
gap> SetPower(coll, 2, F.6*F.8*F.10);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);