Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8813 |
| commutator | C82 | C22×C4 |
| Frattini, cosocle | C2×C82 | C23 |
| prime radical | C23 | #1705 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
391 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 181 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:
(2,6,7,12)(3,9)(4,11)(8,15,14,13)(10,16)(17,18)(19,27,28,39)(20,23)(21,33)(22,35)(24,29,36,41)(25,34)(26,32)(30,44,42,40)(31,45)(37,46)(38,48,47,43);(1,2,3,6,4,8,9,13,5,7,10,12,11,14,16,15)(18,23)(24,36)(25,37)(26,35)(27,39)(33,45)(38,47)(40,44);(17,19,21,29,22,30,34,43,20,28,31,41,32,42,46,48)(18,24,25,27,26,38,33,40,23,36,37,39,35,47,45,44)Polycyclic presentation:
2^10|a^2=d,b^2=ej,c^2=f,e^2=g,f^2=h,g^2=i,h^2=j,b^a=be,c^a=cf,d^b=di,d^c=dj,e^a=eg,f^a=fh,g^a=gi,h^a=hj124 conjugacy classes — view the full character table.
| maximal subgroup of | #72225168, #72226948, #72226953 |
| maximal normal subgroup of | #72225168, #72226948, #72226953 |
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.5*F.10);;
gap> SetPower(coll, 3, F.6);;
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, 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, 6, 1, F.6*F.8);;
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);