Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #111516 |
| commutator | C22×#41 | C22×C4 |
| Frattini, cosocle | C23×#41 | C23 |
| prime radical | #6431743 | C2 |
| LCS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
4479 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 4479 conjugacy classes of subgroups — too many to draw. Only its 183 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:
(1,2)(3,11)(4,16,17,9)(5,20)(6,7)(8,13,23,18)(10,21)(12,28,30,25)(14,32)(15,26)(19,27,31,29)(22,24);(2,7)(3,12,15,30)(4,17)(5,21)(8,24)(9,16)(11,25)(13,14,18,22)(23,32)(26,28);(1,3,5,22,6,15,21,14)(2,8,10,28,7,23,20,25)(4,18,19,30,17,13,31,12)(9,26,27,24,16,11,29,32)Polycyclic presentation:
2^10|a^2=f,b^2=fgij,c^2=ei,e^2=j,b^a=bg,c^a=cd,c^b=cdej,d^a=dh,d^b=dhj,d^c=dj,e^a=ej,e^b=ej,e^c=ej,f^b=fi,f^c=fh,f^d=fj,g^a=gi,g^b=gi,g^d=gj,h^a=hj,i^c=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72240535, #72258337, #72258453, … |
| maximal normal subgroup of | #72240535, #72258337, #72258453, … |
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.6);;
gap> SetPower(coll, 2, F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 3, F.5*F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);