Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #14715 |
| commutator, Frattini, cosocle | C26 | C24 |
| LCS | G ▷ C26 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
12623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(3,7)(5,8)(12,19)(14,20)(15,22)(17,23);(5,8)(9,10)(11,14,18,20)(12,15)(13,16)(17,21,23,24)(19,22);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14,16,20)(12,17)(13,18)(15,21,22,24)(19,23);(1,3,4,7)(2,5,6,8)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24)Polycyclic presentation:
2^10|b^2=g,c^2=e,d^2=ij,b^a=bg,c^a=ce,c^b=cf,d^a=dij,d^b=dh,d^c=dj,e^b=ei,f^b=fi,f^c=fi,f^d=fj,g^c=gi,h^c=hj82 conjugacy classes — view the full character table.
| maximal subgroup of | #72231836, #72236522, #72238051, … |
| maximal normal subgroup of | #72231836, #72236522, #72238051, … |
| maximal quotient of | #72236629, #72263432 |
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);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);