Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#2177 |
| commutator | C2×C16 | C23×C4 |
| Frattini, cosocle | C4×C16 | C24 |
| prime radical | #13697 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×#1311 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2,6,9)(3,10,5,7)(4,8,13,15)(11,16,12,14)(17,18)(19,31)(20,38)(21,42)(22,46)(23,49)(24,30)(25,34)(26,40)(27,32)(28,44)(29,47)(33,57)(35,59)(36,70)(37,71)(39,56)(41,74)(43,54)(45,77)(48,78)(50,63)(51,66)(52,68)(53,73)(55,61)(58,76)(60,67)(62,72)(64,75)(65,79)(69,80);(17,19,32,61,22,36,63,40,47,20,33,62,76,41,43,68,24,35,39,65,48,69,45,66,23,37,64,73,44,67,21,34)(18,25,42,60,28,53,75,71,49,51,77,80,78,79,56,59,30,52,54,74,58,72,57,38,29,26,50,70,46,55,27,31);(1,3,6,5)(2,7,9,10)(4,11,13,12)(8,14,15,16)(17,20,39,73,22,41,45,34,47,35,64,61,76,69,21,40,24,37,32,62,48,67,63,68,23,19,33,65,44,36,43,66)(18,26,54,80,28,55,57,59,49,25,50,74,78,53,27,38,30,51,42,70,58,79,75,31,29,52,77,60,46,72,56,71);(1,4,5,12,6,13,3,11)(2,8,7,14,9,15,10,16)(19,20,35,37)(21,43)(25,51,52,26)(27,56)(31,38,59,71)(32,39)(33,64)(34,66,68,40)(36,41,69,67)(42,54)(45,63)(50,77)(53,79,72,55)(57,75)(60,70,74,80)(61,73,65,62)Polycyclic presentation:
2^10|a^2=h,b^2=eg,c^2=eghj,d^2=f,e^2=gi,f^2=h,g^2=ij,i^2=j,b^a=be,c^a=ceh,d^b=di,d^c=di,e^a=eg,e^d=ej,f^b=fj,f^c=fj,g^a=gi,i^a=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72232026, #72234391, #72260494, … |
| maximal normal subgroup of | #72232026, #72234391, #72260494, … |
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, 1, F.8);;
gap> SetPower(coll, 2, F.5*F.7);;
gap> SetPower(coll, 3, F.5*F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);