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 | #7623625 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ #20495 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,9)(8,12)(10,13)(11,14)(15,22)(16,23)(17,24)(18,26)(19,27)(20,28)(21,29)(25,31)(30,35)(32,34)(33,36);(1,2,4,3)(5,7,9,6)(8,18)(10,25,23,28)(11,14,24,17)(12,21)(13,20,16,31)(15,26)(19,32)(22,29)(27,36)(30,34)(33,35);(2,3)(6,7)(12,15)(13,16)(14,17)(18,29)(25,28)(27,30)(32,33);(1,3,4,2)(8,19,22,35)(10,23)(12,27,15,30)(13,16)(18,32,29,33)(20,31)(21,36,26,34)(25,28);(5,8,20,36,24,35,10,26,9,22,31,34,11,19,23,21)(6,12,28,33,17,30,13,18,7,15,25,32,14,27,16,29)Polycyclic presentation:
2^10|a^2=g,b^2=f,d^2=fj,e^2=fghij,h^2=i,i^2=j,b^a=bj,c^a=cgj,c^b=cgj,d^c=dg,e^b=eh,e^d=ei,f^e=fj,h^b=hi,h^d=hj,h^e=hj,i^b=ij115 conjugacy classes — view the full character table.
| maximal subgroup of | #72314098, #72314408, #72314497, … |
| maximal normal subgroup of | #72314098, #72314408, #72314497, … |
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.6);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);