Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
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 ◁ C2×#2932 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,8,15,10,12,25,28,11,17,24,26,9,16,20,13)(7,14,21,31,22,27,33,36,23,30,34,35,18,32,19,29);(6,12,17,16)(7,18,23,22)(8,24)(13,26,28,15)(14,30)(19,21,33,34)(20,25)(27,32);(2,4)(6,13,17,28)(7,19,23,33)(8,25)(9,10)(12,26,16,15)(14,27)(18,21,22,34)(20,24)(30,32)(31,35);(2,4)(5,7,20,19,9,18,24,34,11,23,25,33,10,22,8,21)(6,14,13,29,16,32,26,35,17,30,28,36,12,27,15,31)Polycyclic presentation:
2^10|a^2=e,b^2=h,c^2=h,d^2=efij,e^2=i,g^2=j,i^2=j,b^a=bg,c^a=cehj,d^a=df,d^b=dgh,d^c=defgh,e^b=ej,e^c=ei,g^c=gj,g^d=gj,h^a=hj,h^d=hj,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72250127, #72250414, #72257162, … |
| maximal normal subgroup of | #72250127, #72250414, #72257162, … |
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.5);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.5*F.6*F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);