Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #245225 |
| commutator, Frattini, cosocle | C23×D4 | C24 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
4442 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 4442 conjugacy classes of subgroups — too many to draw. Only its 157 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 — 4 generators on 32 points:
(1,2)(3,11)(4,8)(5,17)(6,10)(7,13)(9,18)(12,26)(14,24)(15,21)(16,23)(19,28)(20,29)(22,30)(25,27)(31,32);(1,3)(2,7,10,20)(4,12)(5,13)(6,14)(8,19,21,31)(9,11,17,24)(15,27)(16,28)(18,29)(22,26,23,25)(30,32);(4,15)(8,21)(11,24)(12,27)(13,29)(16,30)(19,31)(22,23);(1,4,6,15)(2,8,10,21)(3,12,14,27)(5,16,18,30)(7,19,20,31)(9,23,17,22)(11,25,24,26)(13,28,29,32)Polycyclic presentation:
2^10|b^2=i,d^2=j,e^2=j,b^a=be,c^a=ch,c^b=cf,d^a=dfg,d^c=dj,e^a=ej,e^c=eh,e^d=ei,f^a=fh,f^e=fj,g^a=gh,g^b=gij,h^b=hj,i^a=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72237381, #72238381, #72238382, … |
| maximal normal subgroup of | #72237381, #72238381, #72238382, … |
| maximal quotient of | #72251145, #72251155, #72337668, … |
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, 2, F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);