Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #30396 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | #498878 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20438 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,11,15,27)(8,22)(12,24,23,26)(13,29)(14,31,17,25)(19,34)(28,36,32,35)(30,33);(2,4)(6,12,15,23)(7,16)(8,13)(9,18)(11,24,27,26)(14,32,17,28)(19,33)(22,29)(25,36,31,35)(30,34);(5,6,13,24,18,27,8,23,10,15,29,26,9,11,22,12)(7,17,30,36,20,25,19,28,16,14,33,35,21,31,34,32);(1,2,3,4)(5,7,18,20,10,16,9,21)(6,14,27,31,15,17,11,25)(8,19,29,33,22,34,13,30)(12,28,24,35,23,32,26,36)Polycyclic presentation:
2^10|a^2=h,b^2=h,c^2=ehi,d^2=fgj,e^2=i,g^2=j,i^2=j,c^a=cg,c^b=ce,d^a=dh,d^b=df,d^c=dj,e^a=ej,e^b=ei,e^c=ej,f^d=fj,g^b=gj,g^d=gj,h^c=hj,i^b=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72250127, #72250414, #72250966, … |
| maximal normal subgroup of | #72250127, #72250414, #72250966, … |
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.8);;
gap> SetPower(coll, 3, F.5*F.8*F.9);;
gap> SetPower(coll, 4, F.6*F.7*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*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);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);