Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139848 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C2×#535 | C23 |
| prime radical | #35272 | C22 |
| LCS | G ▷ C23×D4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35272 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1066 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 75 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)(3,12,16,25)(4,18)(5,10)(6,11,21,9)(7,15,24,17)(8,19)(13,28,30,23)(14,29,31,22)(20,27,32,26);(1,3)(2,7,10,25)(4,13)(5,15)(6,17)(8,22,18,23)(9,24,11,12)(14,19)(16,21)(20,31)(26,28,27,29)(30,32);(3,13,15,14)(7,22,25,23)(12,28,24,29)(16,30,17,31)Polycyclic presentation:
2^10|a^2=f,b^2=gi,c^2=gj,d^2=j,h^2=j,b^a=be,c^b=cd,d^a=dg,d^b=dj,e^a=eghj,e^b=ej,e^c=egj,e^d=ej,f^b=fgh,f^e=fij,g^b=gj,h^a=hi,h^e=hj,h^f=hj,i^a=ij,i^b=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72319109, #72319146, #72319302, … |
| maximal normal subgroup of | #72319109, #72319146, #72319302, … |
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.7*F.9);;
gap> SetPower(coll, 3, F.7*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);