Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #30396 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#33541 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,12,15,16)(8,21)(13,27,23,29)(14,31,18,28)(19,24)(22,25)(30,36,32,35)(33,34);(2,4)(6,13)(7,17)(8,22)(9,10)(12,27)(14,30)(15,23)(16,29)(18,32)(19,33)(21,25)(24,34)(28,35)(31,36);(5,6,8,23,9,12,25,27,11,15,21,13,10,16,22,29)(7,18,24,30,26,31,34,36,17,14,19,32,20,28,33,35);(1,2,3,4)(5,7,10,20,11,17,9,26)(6,14,16,31,15,18,12,28)(8,24,22,33,21,19,25,34)(13,30,27,35,23,32,29,36)Polycyclic presentation:
2^10|a^2=h,c^2=eghi,d^2=fhi,e^2=ij,g^2=j,i^2=j,c^a=cg,c^b=ce,d^a=df,d^b=dfgi,d^c=dj,e^a=ej,e^b=ei,e^c=ej,e^d=ej,f^c=fj,g^b=gj,g^d=gj,h^c=hj,i^b=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72250127, #72250414, #72257268, … |
| maximal normal subgroup of | #72250127, #72250414, #72257268, … |
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.5*F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.6*F.8*F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
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.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.9);;
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, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, 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);