Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #110180 |
| commutator | C25 | C2×C42 |
| Frattini, cosocle | C22×H(5,𝔽2) | C23 |
| prime radical | #35346 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
1872 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 1872 conjugacy classes of subgroups — too many to draw. Only its 151 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)(5,19)(8,21)(9,25)(10,15)(11,28)(17,20)(22,27)(24,31)(26,32);(1,2,5,9)(3,11,13,26)(4,15,17,25)(6,10,19,27)(7,20,22,18)(8,23,24,14)(12,28,29,32)(16,21,30,31);(1,3)(2,8,10,11)(4,16,18,12)(5,13,19,23)(6,14)(7,21)(9,26)(15,28)(17,30)(20,29)(22,32,25,31)(24,27)Polycyclic presentation:
2^10|b^2=e,c^2=g,b^a=bd,c^a=cf,c^b=cfij,d^b=dh,d^c=dhi,e^a=eh,e^c=ehij,f^b=fhi,f^c=fij,f^e=fj,g^a=gij,g^b=gij,g^d=gj,g^e=gj,h^c=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72265989, #72285566, #72285612, … |
| maximal normal subgroup of | #72265989, #72285566, #72285612, … |
| maximal quotient of | #72265909 |
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.5);;
gap> SetPower(coll, 3, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, 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);