Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#535 |
| commutator, Frattini, cosocle | C23×C4 | C25 |
| socle | C22 | #34963 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,12)(8,19)(9,11)(10,14)(13,22)(15,21)(16,29)(17,25)(18,23)(20,24)(26,32)(27,31)(28,33)(30,34)(35,36);(12,25)(23,27)(26,35)(29,34);(2,4)(6,11,14,15)(8,20)(12,23,25,27)(13,19,28,24)(16,30)(22,33)(26,29,35,34)(32,36);(1,2,3,4)(5,7)(6,12)(8,16)(9,18)(10,17)(11,23)(13,26)(14,25)(15,27)(19,29)(20,30)(21,31)(22,32)(24,34)(28,35)(33,36);(5,8,21,33,10,20,9,22)(6,13,11,24,14,28,15,19)(7,16,31,36,17,30,18,32)(12,26,23,34,25,35,27,29)Polycyclic presentation:
2^10|c^2=h,d^2=f,e^2=ij,i^2=j,b^a=bg,c^a=ci,d^a=df,d^b=dh,d^c=df,e^a=ei,e^c=ej,g^d=gj,h^a=hj,i^a=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72249989, #72250233, #72256934, … |
| maximal normal subgroup of | #72249989, #72250233, #72256934, … |
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, 3, F.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);