Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #12067 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | #1311 | C23 |
| prime radical | C2×D16 | C2×C8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ #7053 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
182 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,12,13)(11,15,16,14)(18,26)(19,29)(20,37)(21,42)(22,38)(23,24)(27,33)(28,45)(30,36)(31,35)(32,47)(34,44)(39,40)(41,48)(43,46);(1,3,4,11,6,9,13,15,5,10,12,16,2,7,8,14)(17,18,20,19,21,27,39,33,23,29,40,26,22,44,37,45,25,31,32,36,38,46,48,43,24,30,41,35,42,28,47,34);(3,10)(7,9)(11,16)(14,15)(17,19,32,44,38,33,39,31,23,26,41,46,42,45,37,30,25,36,20,28,21,43,48,18,24,35,40,27,22,34,47,29)Polycyclic presentation:
2^10|a^2=hi,b^2=d,c^2=ehij,d^2=f,e^2=gij,f^2=h,g^2=i,h^2=j,i^2=j,b^a=be,c^a=ceghij,c^b=cghi,d^a=dg,d^c=di,e^a=eg,e^b=ei,e^d=ej,f^a=fi,f^c=fj,g^a=gij,g^b=gj,h^a=hj,i^a=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72287210, #72287265, #72287277, … |
| maximal normal subgroup of | #72287210, #72287265, #72287277, … |
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*F.9);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);