Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #23708 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#13323 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,11,14,27)(8,20)(10,24)(12,25,22,15)(13,29,30,26)(17,34,31,28)(18,35)(19,33)(21,23)(32,36);(2,4)(6,12)(7,16)(8,21)(10,24)(11,25)(13,28)(14,22)(15,27)(17,29)(18,35)(20,23)(26,31)(30,34)(32,36);(2,4)(5,6,10,25,9,14,24,15)(7,17,19,13,16,31,33,30)(8,22,20,11,23,12,21,27)(18,29,35,34,32,26,36,28);(1,2,3,4)(5,7,10,19,9,16,24,33)(6,13,25,31,14,30,15,17)(8,18,20,35,23,32,21,36)(11,26,12,28,27,29,22,34)Polycyclic presentation:
2^10|a^2=h,c^2=hi,d^2=fghij,e^2=i,i^2=j,c^a=ce,c^b=ceij,d^a=dg,d^b=df,d^c=dfhij,e^a=ei,e^b=eij,e^c=ei,f^c=fj,f^d=fj,g^c=gj,g^d=gj,h^c=hj,i^a=ij,i^b=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72250312, #72257241, #72257311, … |
| maximal normal subgroup of | #72250312, #72257241, #72257311, … |
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, 3, F.8*F.9);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, 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, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);