Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #93099 |
| commutator | C26 | C42 |
| Frattini, prime radical, cosocle | C2×#2613 | C22 |
| LCS | G ▷ C26 ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2613 ◁ G | |
| LECS | G ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
1473 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 1473 conjugacy classes of subgroups — too many to draw. Only its 47 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 — 2 generators on 32 points:
(1,2,4,8)(3,7,11,18)(5,10,16,21)(6,9,15,22)(12,26,27,30)(13,20,14,29)(17,23,24,32)(19,25,31,28);(1,3,5,12)(2,7,10,19)(4,14,16,27)(6,13,17,28)(8,18,21,30)(9,20,23,26)(11,24,25,15)(22,29,32,31)Polycyclic presentation:
2^10|a^2=c,b^2=e,b^a=bd,c^b=cf,d^a=df,d^b=dg,d^c=dj,e^a=eg,e^c=eh,e^d=ei,f^a=fj,f^b=fh,f^e=fj,g^a=gh,g^b=gi,h^b=hj,i^a=ij40 conjugacy classes — view the full character table.
| maximal subgroup of | #72323219, #72323239, #72323240, … |
| maximal normal subgroup of | #72323219, #72323239, #72323240, … |
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, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);