Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #30396 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,17,9,16,20,36,12,15,30,31,11,18,29,28)(7,19,23,35,27,13,26,32,24,34,25,33,22,21,8,14);(1,2,3,4)(5,7)(6,13)(8,20)(9,27)(10,25)(11,22)(12,24)(14,17)(15,21)(16,34)(18,19)(23,30)(26,29)(28,33)(31,32)(35,36);(5,8,9,23,12,26,11,25)(6,14,16,35,15,32,18,33)(7,20,27,30,24,29,22,10)(13,31,34,28,21,17,19,36);(2,4)(5,6)(7,21)(8,14)(9,18)(10,28)(11,16)(12,15)(13,24)(17,29)(19,22)(20,31)(23,33)(25,35)(26,32)(27,34)(30,36)Polycyclic presentation:
2^10|a^2=h,b^2=f,c^2=ij,g^2=j,h^2=ij,i^2=j,b^a=bg,c^b=ce,d^a=dh,d^b=dfj,d^c=dgij,e^a=ej,e^d=ej,g^b=gj,g^c=gj,g^d=gj,h^b=hj,h^d=hi,i^d=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72321276, #72321747, #72322118, … |
| maximal normal subgroup of | #72321276, #72321747, #72322118, … |
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.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);