Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
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 | |
1556 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1556 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:
(1,2,5,10)(3,11,13,27,14,7,19,22)(4,15,18,29)(6,9,20,26)(8,23,25,17)(12,24,28,32,16,21,30,31);(3,12,14,16)(7,21,11,24)(13,28,19,30)(22,31,27,32);(1,3)(2,7)(4,16)(5,19)(6,14)(8,24)(9,11)(10,22)(12,17)(13,20)(15,21)(18,28)(23,30)(25,32)(26,27)(29,31)Polycyclic presentation:
2^10|a^2=f,b^2=hj,d^2=j,f^2=hj,c^a=ce,c^b=cd,d^a=dh,d^c=dj,e^a=eg,e^b=ehj,e^d=ej,f^c=fg,f^e=fi,g^a=gi,g^f=gj,h^c=hj,i^a=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72285086, #72285887, #72285974, … |
| maximal normal subgroup of | #72285086, #72285887, #72285974, … |
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.8*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);