Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#442 |
| commutator | C2×C8 | C23×C8 |
| Frattini, cosocle | C82 | C24 |
| socle | C22 | C2×#1580 |
| prime radical | #33660 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #11402 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(17,18)(19,24)(20,26)(21,27)(22,25)(23,28)(29,36)(30,37)(31,40)(32,38)(33,41)(34,39)(35,42)(43,46)(44,47)(45,48);(2,3)(4,16)(5,13)(6,7)(9,11)(10,14)(12,15)(17,19,23,22)(18,24,28,25)(20,29,33,32)(21,30,35,34)(26,36,41,38)(27,37,42,39)(31,43,45,44)(40,46,48,47);(1,2)(3,13)(4,15)(5,9)(6,11)(7,10)(8,12)(14,16)(17,19,23,22)(18,25,28,24)(20,29,33,32)(21,30,35,34)(26,38,41,36)(27,39,42,37)(31,43,45,44)(40,47,48,46);(1,3,4,2,6,9,5,10,8,15,13,12,7,14,16,11)(17,20,21,31,22,32,34,44,23,33,35,45,19,29,30,43)(18,26,27,40,25,38,39,47,28,41,42,48,24,36,37,46)Polycyclic presentation:
2^10|b^2=ij,c^2=ij,d^2=e,e^2=g,f^2=h,g^2=i,h^2=j,c^a=cij,c^b=cf,d^b=dfhj,d^c=df,e^b=eh,e^c=ehj,f^b=fhj,f^c=fhj,f^d=fj,g^b=gj,g^c=gj,h^b=hj,h^c=hj124 conjugacy classes — view the full character table.
| maximal subgroup of | #72286736, #72320302, #72320884, … |
| maximal normal subgroup of | #72286736, #72320302, #72320884, … |
| maximal quotient of | #72287277, #72287300, #72287377, … |
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.9*F.10);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);