Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #9837 |
| commutator | C82 | C22×C4 |
| Frattini, cosocle | C2×C82 | C23 |
| prime radical | #19603 | C22 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
556 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 137 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,8,10)(3,7)(4,12)(9,14,16,13)(11,15)(18,22,25,29)(19,26)(20,27)(23,28,32,30)(24,31);(1,2,7,14,12,16,3,10,5,8,15,13,4,9,11,6)(18,23,25,32)(19,24)(22,28,29,30)(26,31);(17,18,24,29,20,23,31,28,21,25,19,22,27,32,26,30)Polycyclic presentation:
2^10|a^2=d,b^2=deg,c^2=fj,e^2=g,f^2=h,g^2=i,h^2=j,b^a=be,c^a=cf,c^b=ch,d^b=di,d^c=dj,e^a=eg,e^b=ei,f^a=fh,f^b=fj,g^a=gi,h^a=hj76 conjugacy classes — view the full character table.
| maximal subgroup of | #72225261, #72225309, #72226946, … |
| maximal normal subgroup of | #72225261, #72225309, #72226946, … |
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, 2, F.4*F.5*F.7);;
gap> SetPower(coll, 3, F.6*F.10);;
gap> SetPower(coll, 5, F.7);;
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.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
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.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);