Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C8 | C23×C42 |
| Frattini, cosocle | C22×C8 | C25 |
| prime radical | C22×#509 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ #20331 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
4457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,15,17,28)(8,18)(9,20)(11,21,23,33)(13,26)(14,25)(16,27,30,29)(19,31)(22,32,35,34)(24,36);(7,16,17,30)(8,19)(11,22,23,35)(13,24)(15,27,28,29)(18,31)(21,32,33,34)(26,36);(1,2,3,4)(5,6)(7,11)(8,13)(9,14)(10,12)(15,21)(16,22)(17,23)(18,26)(19,24)(20,25)(27,32)(28,33)(29,34)(30,35)(31,36);(5,7,8,15,9,16,18,29,10,17,19,28,20,30,31,27)(6,11,13,21,14,22,26,34,12,23,24,33,25,35,36,32);(6,12)(11,23)(13,24)(14,25)(21,33)(22,35)(26,36)(32,34)Polycyclic presentation:
2^10|a^2=g,b^2=g,c^2=f,d^2=h,h^2=i,i^2=j,d^a=dh,d^b=di,e^c=ej,g^d=gj,h^a=hi,h^b=hj,i^a=ij178 conjugacy classes — view the full character table.
| maximal subgroup of | #72239561, #72250849, #72250936, … |
| maximal normal subgroup of | #72239561, #72250849, #72250936, … |
| maximal quotient of | #72319356, #72320051, C2×#72239482 |
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.7);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*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.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);