Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8534 |
| commutator | C22×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×#121 | C23 |
| prime radical | #4598732 | C2 |
| LCS | G ▷ C22×C4⋊C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×#121 ▷ C24 ▷ C22 ▷ C1 | |
2744 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 2744 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(1,2,4,10,8,13,3,12)(5,20,23,29,25,9,22,26)(6,14,18,30,7,15,16,31)(11,19,32,24,28,17,27,21);(2,9,13,20)(3,16)(4,18)(5,21)(10,29,12,26)(11,15,28,14)(17,19)(22,23)(24,25)(27,31,32,30);(1,3)(4,8)(5,22)(6,16)(7,18)(9,11)(10,12)(17,21)(19,24)(20,28)(23,25)(26,27)(29,32)(30,31)Polycyclic presentation:
2^10|a^2=d,b^2=hi,d^2=j,e^2=ij,f^2=j,b^a=be,c^a=cf,c^b=cij,d^b=dg,d^c=dj,e^a=egij,e^b=ei,e^c=eh,e^d=ei,f^b=fhj,f^c=fj,f^e=fj,g^a=gi,h^a=hj52 conjugacy classes — view the full character table.
| maximal subgroup of | #72238694 |
| maximal normal subgroup of | #72238694 |
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.4);;
gap> SetPower(coll, 2, F.8*F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
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*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);