Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #1333 |
| commutator | C16 | C22×C16 |
| Frattini, cosocle | #1311 | C23 |
| socle | C22 | C2×#1333 |
| prime radical | C22×D16 | C8 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ C2×#1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,8,5,9,12,15,6,10,13,16,3,7,11,14)(17,18,19,25,21,27,32,28,22,23,33,41,39,29,30,43)(20,34,36,26,37,24,35,44,38,45,31,42,47,40,48,46);(18,23)(24,40)(25,41)(26,42)(27,29)(28,43)(34,45)(44,46);(1,3,6,5)(2,7,10,9)(4,11,13,12)(8,14,16,15)(18,24)(19,30)(20,35)(21,39)(23,40)(25,42)(26,41)(27,34)(28,44)(29,45)(31,47)(32,33)(36,37)(38,48)(43,46)Polycyclic presentation:
2^10|a^2=d,c^2=hj,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,b^a=bj,c^a=ce,d^c=dij,e^a=eg,e^c=egij,e^d=ej,f^c=fj,g^a=gi,g^c=gij,i^a=ij,i^c=ij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72286865, #72287236, #72287393, … |
| maximal normal subgroup of | #72286865, #72287236, #72287393, … |
| maximal quotient of | #72287258 |
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.4);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);