Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C25 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
5910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(18,22)(19,25)(20,27)(23,29)(26,32)(30,31);(1,3,6,2)(4,7,12,9)(5,10,14,8)(11,15,13,16)(17,18)(19,23)(20,28)(21,22)(24,27)(25,29)(26,30)(31,32);(2,3)(7,9)(8,10)(15,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,31)(27,32)(28,30);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,24)(19,26,25,32)(21,27)(22,28)(23,30,29,31);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(19,23)(20,27)(24,28)(25,29)(26,31)(30,32)Polycyclic presentation:
2^10|b^2=fi,d^2=fj,e^2=fi,b^a=bf,c^a=cfh,c^b=cfi,d^a=di,d^b=df,d^c=dhi,e^c=eg,e^d=efij,f^c=fj,g^a=gi,g^d=gi,h^b=hj,h^c=hj,h^d=hj,h^e=hij,h^g=hj,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72312232, #72312260, #72312560, … |
| maximal normal subgroup of | #72312232, #72312260, #72312560, … |
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.6*F.9);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.6*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);