Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#12696 |
| commutator | C2×C42 | C23×C4 |
| Frattini, cosocle | #532 | C24 |
| prime radical | #10575552 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ C2×#3498 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
4650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,8)(5,10)(6,11)(7,14)(9,23)(13,29)(15,26)(16,25)(17,24)(18,21)(19,20)(22,28)(27,31)(30,32);(1,3,2,7)(4,17,19,24)(5,15,9,25)(6,16,11,26)(8,27,20,13)(10,12,23,14)(18,32,22,30)(21,31,28,29);(1,4,5,21,6,22,23,20)(2,8,10,18,11,28,9,19)(3,13,14,30,16,31,15,17)(7,24,25,29,26,32,12,27);(1,2)(3,7)(4,18)(5,9)(6,11)(8,21)(10,23)(12,14)(13,29)(15,25)(16,26)(17,32)(19,22)(20,28)(24,30)(27,31)Polycyclic presentation:
2^10|b^2=e,c^2=h,f^2=j,g^2=i,h^2=j,b^a=bf,c^a=ci,c^b=cg,d^c=dhj,e^c=eh,f^a=fj,f^b=fj,f^c=fj,g^b=ghij,g^d=gi,g^e=gi,h^b=hi,h^d=hj,h^e=hj,i^b=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72339824, #72339838, #72339850, … |
| maximal normal subgroup of | #72339824, #72339838, #72339850, … |
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.5);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);