Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #106654 |
| commutator | C23×C4 | C2×C42 |
| Frattini, cosocle | C22×H(5,𝔽2) | C23 |
| prime radical | C2×#3498 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,9)(5,10,14,7)(6,11)(8,15)(13,25)(16,27,19,32)(17,26)(18,24,29,21)(20,31,22,30)(23,28);(2,7)(3,13)(4,17)(5,14)(8,22)(9,24)(10,11)(12,27)(20,23)(21,26)(25,32)(30,31);(1,3,14,16,6,13,5,19)(2,8,10,22,11,23,7,20)(4,15,18,31,17,28,29,30)(9,25,21,32,26,12,24,27)Polycyclic presentation:
2^10|a^2=f,c^2=eg,d^2=j,e^2=j,b^a=be,c^a=cd,c^b=ci,d^a=di,d^b=dj,d^c=dh,e^a=ehi,e^b=ej,e^c=ej,f^b=fhij,f^c=fij,f^e=fj,g^a=gij,g^c=gj,g^d=gj,h^a=hj,h^c=hj,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72241334, #72241336, #72241340, … |
| maximal normal subgroup of | #72241334, #72241336, #72241340, … |
| maximal quotient of | C2×#72241333 |
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.6);;
gap> SetPower(coll, 3, F.5*F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);