Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #13674 |
| commutator | #492 | C22×C4 |
| Frattini, cosocle | C2×#492 | C23 |
| prime radical | C2×#1624 | C22 |
| LCS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#492 ◁ G | |
| LECS | G ▷ C2×#492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1070 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,17,26,29,20,28,19,12)(8,21)(9,14)(10,15)(11,16)(13,23)(18,33,34,36,35,30,22,27)(24,32)(25,31);(7,18,26,34,20,35,19,22)(12,27,17,33,29,36,28,30);(1,2,3,4)(5,7)(6,12)(8,22)(9,19)(10,26)(11,20)(13,30)(14,17)(15,28)(16,29)(18,24)(21,36)(23,35)(25,34)(27,31)(32,33)Polycyclic presentation:
2^10|a^2=ghij,b^2=ghij,c^2=d,e^2=hj,g^2=i,h^2=j,c^a=cf,c^b=ce,e^a=eg,e^c=eh,f^a=fhj,f^b=fgh,f^e=fh,g^a=gi,g^c=gh,g^f=gi,h^a=hi,h^c=hj,h^f=hj,i^c=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72283544, #72283855, #72283945 |
| maximal normal subgroup of | #72283544, #72283855, #72283945 |
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*F.8*F.9*F.10);;
gap> SetPower(coll, 2, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);