Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #7112 |
| commutator | C4×C16 | C2×C8 |
| Frattini, cosocle | C4×#346 | C22 |
| prime radical | #33733 | C4 |
| LCS | G ▷ C4×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ C4×#346 ◁ G | |
| LECS | G ▷ C4×#346 ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
284 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 128 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,11,12,15,13,16,14,7)(4,8)(5,9)(6,10)(17,18)(19,28,30,23)(20,33)(21,36)(22,26)(24,29,41,44)(25,34)(27,37)(31,40)(32,39,46,43)(35,48)(38,45,42,47);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15)(18,23)(19,29)(20,31)(21,37)(24,40)(25,41)(26,28)(27,43)(30,44)(32,47)(33,42)(34,35)(36,39)(38,48)(45,46)Polycyclic presentation:
2^10|a^2=c,c^2=e,d^2=fghij,f^2=h,g^2=i,h^2=j,b^a=bd,c^b=cg,d^a=df,d^b=dfg,d^c=dj,e^b=ei,f^a=fh,f^b=fhj,g^a=gj,g^b=gi,h^a=hj,h^b=hj94 conjugacy classes — view the full character table.
| maximal subgroup of | #72227210 |
| maximal normal subgroup of | #72227210 |
| maximal quotient of | #72227206, #72227207 |
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.3);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);