Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8541 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C22×#120 | C23 |
| prime radical | #35346 | C22 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
2937 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 2937 conjugacy classes of subgroups — too many to draw. Only its 105 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:
(1,2,5,10)(3,7,13,20)(4,8,14,22)(6,9,16,23)(11,19,24,18)(12,17,26,30)(15,21,27,31)(25,29,32,28);(2,7,8,18)(3,11,13,24)(4,14)(9,17,21,28)(10,20,22,19)(12,25,26,32)(15,27)(23,30,31,29);(3,12)(4,15)(7,17)(8,21)(13,26)(14,27)(20,30)(22,31)Polycyclic presentation:
2^10|a^2=i,b^2=e,b^a=bd,c^b=cf,d^b=dg,d^c=dj,e^a=eg,e^c=eh,e^d=ei,f^b=fh,f^d=fj,g^b=gi,g^c=gj,h^a=hj64 conjugacy classes — view the full character table.
| maximal subgroup of | #72309392, #72310514, #72310529, … |
| maximal normal subgroup of | #72309392, #72310514, #72310529, … |
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.9);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
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);;
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);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);