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 | #24089 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#3058 | C22 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2211 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)(6,12,16,25)(7,17,20,21)(8,22)(13,26,15,29)(14,30)(18,33,19,34)(23,24)(27,32);(1,2)(3,4)(5,6,8,15,10,12,23,26,11,16,22,13,9,25,24,29)(7,14,19,31,21,27,34,35,20,30,18,28,17,32,33,36);(6,13,16,15)(7,18,20,19)(8,23)(9,10)(12,26,25,29)(14,27)(17,33,21,34)(22,24)(28,31)(30,32);(2,4)(5,7,8,19,10,21,23,34,11,20,22,18,9,17,24,33)(6,14,15,31,12,27,26,35,16,30,13,28,25,32,29,36)Polycyclic presentation:
2^10|a^2=g,b^2=e,c^2=g,d^2=e,e^2=i,f^2=j,h^2=j,i^2=j,b^a=bh,c^b=cei,d^a=df,d^b=dfgh,d^c=dej,e^a=ej,e^c=ei,f^b=fj,f^c=fj,g^b=gj,g^d=gj,h^c=hj,h^d=hj,i^c=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72250334, #72252629, #72252751, … |
| maximal normal subgroup of | #72250334, #72252629, #72252751, … |
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, 2, F.5);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*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, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);