Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C217 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#35346 |
| commutator, Frattini, cosocle | C25 | C26 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
351962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 20 points:
(11,15)(19,20);(5,6)(7,10)(8,15,17,11)(9,12)(13,18)(14,20,16,19);(5,7)(6,10)(8,16)(9,13)(11,20)(12,18)(14,17)(15,19);(7,13)(10,18);(1,2)(3,4)(5,8)(6,11)(7,14)(9,17)(10,19)(12,15)(13,16)(18,20);(2,4)Polycyclic presentation:
2^11|b^2=h,b^a=bh,d^c=dhk,e^a=ei,e^b=ei,e^c=ek,e^d=eg,f^e=fj,g^c=gk,h^e=hk,i^b=ik212 conjugacy classes — view the full character table.
| maximal subgroup of | #72258990, #72259085, #72264871, … |
| maximal normal subgroup of | #72258990, #72259085, #72264871, … |
| maximal quotient of | #72265593 |
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.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);