Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #5021210 |
| commutator, Frattini, cosocle | C23×C4 | C25 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,11)(3,12)(4,9,19,28)(5,10,21,8)(7,15)(13,24)(14,29)(16,22)(17,32)(18,27,20,25)(23,31)(26,30);(1,3)(2,7,11,24)(4,17)(5,13)(6,16)(8,22,10,12)(9,26,28,29)(14,20)(15,21)(18,30)(19,31)(23,25,32,27);(3,13,16,15)(7,12,24,22)(14,17,30,31)(23,29,32,26);(2,8)(5,21)(7,22)(9,27)(10,11)(12,24)(13,15)(14,30)(18,20)(23,26)(25,28)(29,32);(1,4)(2,9)(3,14)(5,18)(6,19)(7,23)(8,25)(10,27)(11,28)(12,29)(13,17)(15,31)(16,30)(20,21)(22,26)(24,32)Polycyclic presentation:
2^10|a^2=g,b^2=gi,c^2=gj,f^2=j,h^2=j,b^a=bghj,c^b=cf,d^a=dghij,d^c=dgj,e^b=eghi,e^d=eij,f^b=fj,f^d=fj,g^b=gj,h^b=hj,h^d=hj,h^e=hj,i^a=ij,i^b=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72319109, #72319501, #72338567, … |
| maximal normal subgroup of | #72319109, #72319501, #72338567, … |
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.7*F.9);;
gap> SetPower(coll, 3, F.7*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);