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 | #139908 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | #2309 | C23 |
| prime radical | #417566 | C2 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35272 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1555 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1555 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:
(3,11,14,18)(7,20,12,25)(13,26,15,28)(21,30,27,31);(1,2,4,8)(3,12,13,27,14,7,15,21)(5,10,16,22)(6,9,17,23)(11,20,26,30,18,25,28,31)(19,24,29,32);(1,3)(2,7)(4,15)(5,18)(6,14)(8,21)(9,12)(10,25)(11,19)(13,17)(16,26)(20,24)(22,31)(23,27)(28,29)(30,32)Polycyclic presentation:
2^10|a^2=g,b^2=d,d^2=g,f^2=j,b^a=bg,c^a=cf,c^b=ce,d^c=dh,e^a=eg,e^b=eh,e^d=ei,f^b=fg,f^c=fj,f^e=fj,g^c=gj,h^b=hi,h^d=hj,i^b=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72339016, #72339018, #72339023, … |
| maximal normal subgroup of | #72339016, #72339018, #72339023, … |
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.7);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);