Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #140032 |
| commutator | #379 | C22×C4 |
| Frattini, cosocle | #2309 | C23 |
| prime radical | #485608 | C2 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1093 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 1093 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,11)(8,21)(9,19)(10,20)(18,23)(22,30)(24,31)(26,32)(27,29);(1,2,5,7)(3,8,16,21,11,23,13,18)(4,9,15,19)(6,10,17,20)(12,26,29,30,25,22,27,32)(14,24,28,31);(1,3,4,12,6,11,14,25)(2,8,9,22,10,23,24,26)(5,16,15,29,17,13,28,27)(7,18,19,30,20,21,31,32)Polycyclic presentation:
2^10|b^2=fh,c^2=d,d^2=j,f^2=h,b^a=bf,c^a=ce,c^b=cegj,d^a=dh,d^b=dh,e^b=eg,e^c=eh,e^d=ej,f^a=fh,f^c=fgi,f^e=fij,g^a=gi,g^b=gij,g^f=gj,h^c=hj,i^b=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72284932, #72285936, #72286012, … |
| maximal normal subgroup of | #72284932, #72285936, #72286012, … |
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, 2, F.6*F.8);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);