Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13049 |
| commutator | C4⋊Q8 | C23×C4 |
| Frattini, cosocle | #532 | C24 |
| prime radical | #6348452 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,9,19,12)(3,13,15,29,16,26,32,30)(4,10,18,27,6,20,17,28)(8,22,23,21,25,31,24,14);(1,3,7,16)(2,8,9,25)(4,14,6,21)(5,15,19,32)(10,26,20,13)(11,23,12,24)(17,31,18,22)(27,30,28,29);(2,9)(4,17)(5,19)(6,18)(8,23)(10,27)(13,26)(14,31)(15,32)(20,28)(21,22)(24,25);(1,4)(2,10)(3,14)(5,18)(6,7)(8,13)(9,20)(11,27)(12,28)(15,22)(16,21)(17,19)(23,29)(24,30)(25,26)(31,32)Polycyclic presentation:
2^10|a^2=e,b^2=j,e^2=j,f^2=j,g^2=i,h^2=j,b^a=bf,c^a=ci,c^b=cg,d^b=di,d^c=dh,e^c=ej,f^a=fj,f^b=fj,f^c=fh,f^d=fj,g^a=ghij,g^b=gi,g^c=gi,g^d=gi,g^f=gi,h^b=hi,h^c=hj,h^d=hj,h^f=hj,i^a=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72286520, #72286703, #72286897, … |
| maximal normal subgroup of | #72286520, #72286703, #72286897, … |
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.5);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 5, F.10);;
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, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
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.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);