Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #24248 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20386 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,8,13,9,14,21,30,10,11,19,24,22,25,34,31)(7,15,16,32,18,12,26,28,17,29,20,35,33,23,36,27);(6,11)(8,19)(14,25)(16,20)(21,34)(26,36)(27,28)(32,35);(1,2,3,4)(5,7)(6,12,11,23)(8,20)(9,18)(10,17)(13,27,24,28)(14,29,25,15)(16,19)(21,36)(22,33)(26,34)(30,32,31,35);(2,4)(6,13,11,24)(8,21)(9,22)(12,27,23,28)(14,31,25,30)(15,32,29,35)(16,26)(18,33)(19,34)(20,36)Polycyclic presentation:
2^10|a^2=g,c^2=e,d^2=f,g^2=i,i^2=j,b^a=bh,c^a=cij,c^b=cf,d^a=dgij,d^b=df,d^c=def,e^a=ej,f^a=fj,g^b=gj,g^c=gj,g^d=gi,h^a=hj,h^c=hj,i^d=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72226831 |
| maximal normal subgroup of | #72226831 |
| maximal quotient of | #72224937 |
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.7);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*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.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);