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 | C22 | #13086 |
| commutator | C2×#117 | C22×C4 |
| Frattini, cosocle | #2297 | C23 |
| prime radical | #35527 | C22 |
| LCS | G ▷ C2×#117 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #35527 ◁ G | |
| LECS | G ▷ #2297 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1558 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 1558 conjugacy classes of subgroups — too many to draw. Only its 79 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:
(2,7,10,25,11,27,18,26)(3,13,15,32)(4,17)(8,29,16,20)(9,28,30,14,21,24,12,31)(19,23);(1,2,8,26)(3,14,5,21)(4,18,32,28)(6,11,15,25)(7,23,12,13)(9,16,31,22)(10,29,24,19)(17,30,20,27);(2,9)(4,19)(7,24)(10,12)(11,21)(13,20)(14,25)(17,23)(18,30)(26,31)(27,28)(29,32)Polycyclic presentation:
2^10|a^2=e,b^2=defh,d^2=hi,e^2=h,g^2=j,b^a=bd,c^a=cfh,c^b=chij,d^a=dg,d^b=dgij,d^c=dh,e^b=eghij,e^c=eh,e^d=eij,f^a=fh,f^b=fh,f^d=fj,g^a=gi,g^b=gi,g^c=gj,g^d=gj,h^b=hj,i^b=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72338950, #72338966, #72338968, … |
| maximal normal subgroup of | #72338950, #72338966, #72338968, … |
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.5);;
gap> SetPower(coll, 2, F.4*F.5*F.6*F.8);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, 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, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);