Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C22×C8 | C25 |
| prime radical | #7622978 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ C2×#2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,13)(9,14)(10,12)(15,18)(16,26)(17,27)(19,22)(20,29)(21,28)(23,25)(24,30)(31,36)(32,35)(33,34);(2,4)(5,6)(7,15,17,32)(8,19)(9,23)(10,12)(11,18,27,35)(13,22)(14,25)(16,29,24,33)(20,30,34,26)(21,31)(28,36);(2,4)(5,7)(6,11)(8,20)(9,24)(10,17)(12,27)(13,29)(14,30)(15,31)(16,25)(18,36)(19,32)(21,34)(22,35)(23,26)(28,33);(7,16,17,24)(8,21)(11,26,27,30)(13,28)(15,29,32,33)(18,20,35,34)(19,31)(22,36);(6,12)(7,17)(13,28)(14,23)(16,24)(18,35)(19,31)(20,34)Polycyclic presentation:
2^10|b^2=f,d^2=f,g^2=i,i^2=j,b^a=bh,c^a=ch,c^b=cg,d^c=dij,e^a=ej,e^b=ej,e^c=ej,f^c=fj,g^b=gi,g^c=gij,g^d=gj,i^b=ij,i^c=ij130 conjugacy classes — view the full character table.
| maximal subgroup of | #72320583, #72343466, #72343493, … |
| maximal normal subgroup of | #72320583, #72343466, #72343493, … |
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, 2, F.6);;
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.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*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);