Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152009 |
| commutator | #532 | C22×C4 |
| Frattini, cosocle | #3147 | C23 |
| prime radical | #6353305 | C2 |
| LCS | G ▷ #532 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ C4×#41 ◁ #60647 ◁ G | |
| LECS | G ▷ #3147 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1733 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 1733 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(2,8)(3,12)(4,17)(5,22)(10,26)(13,30)(14,31)(15,28)(16,19)(21,25)(23,29)(24,27);(1,2,9,22,7,8,6,5)(3,13)(4,18,25,20,17,11,21,32)(12,28)(14,23)(15,16)(19,30)(26,27);(1,3,11,26,6,13,20,29,7,16,18,27,9,15,32,31)(2,10,25,30,5,23,4,19,8,24,21,28,22,14,17,12)Polycyclic presentation:
2^10|b^2=ghij,c^2=dehij,d^2=j,e^2=hi,f^2=j,g^2=i,h^2=j,b^a=bj,c^a=cd,c^b=cf,d^a=dj,d^b=dj,d^c=dj,e^a=ej,e^b=eg,e^c=ej,f^a=fj,f^b=fh,f^c=fg,f^e=fhij,g^b=gi,g^c=gh,g^f=gi,h^b=hi,h^c=hj,h^f=hj,i^c=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72283911, #72284014, #72284059, … |
| maximal normal subgroup of | #72283911, #72284014, #72284059, … |
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.7*F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.4*F.5*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*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.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);