Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C222 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C27 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
At least 6518 conjugacy classes of subgroups; only the 47776 normal subgroups are shown.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 20 points:
(13,14)(15,20)(16,18)(17,19);(9,10)(11,12)(13,15,17,18)(14,16,19,20);(9,10)(11,12);(5,6)(7,8)(9,10,11,12)(13,14)(15,16)(17,19)(18,20);(1,2,3,4)(5,6,7,8)(9,10)(11,12);(13,16)(14,18)(15,19)(17,20);(1,2)(3,4)(5,6)(7,8)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^11|b^2=j,d^2=i,e^2=k,b^a=bj,d^b=di,d^c=di,e^d=eh,f^d=fj,g^d=gj,g^e=gk344 conjugacy classes — view the full character table.
| maximal subgroup of | #72256591, C2×#72256589 |
| maximal normal subgroup of | #72256591, C2×#72256589 |
| maximal quotient of | #72330739, D4×#612424, C2×#72256589, … |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);