Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
68029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,24)(21,23);(1,2,3,4)(5,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,20)(18,22,23,24);(1,2,3,4)(6,8)(9,11,12,16)(10,13,15,14)(17,18,21,23)(19,22,20,24);(9,11)(10,14)(12,16)(13,15)(17,20,21,19)(18,22,23,24);(5,6)(7,8);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,24)(20,22)(21,23)Polycyclic presentation:
2^10|b^2=gi,c^2=ghi,d^2=j,b^a=bg,c^a=cgj,c^b=ch,d^a=dj,d^c=di,e^b=ehij,e^c=ehij,f^b=fgj,f^c=fgj172 conjugacy classes — view the full character table.
| maximal subgroup of | #72233717, C2×#72233693 |
| maximal normal subgroup of | #72233717, C2×#72233693 |
| maximal quotient of | #72334416, C2×#72233693, #72335014, … |
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, 2, F.7*F.9);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);