Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #8546 |
| commutator | C22×#41 | C22×C4 |
| Frattini, prime radical, cosocle | C23×#41 | C23 |
| socle | C2 | #122051 |
| LCS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
2143 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 2143 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(3,11,15,16)(7,20)(8,23)(12,27)(14,28,19,31)(21,30)(22,24)(25,32);(1,2,5,9,6,10,13,26)(3,12,14,7,15,21,19,22)(4,8,17,25,18,23,29,32)(11,27,28,20,16,30,31,24);(1,3,13,14,6,15,5,19)(2,7,9,12,10,22,26,21)(4,16,29,31,18,11,17,28)(8,24,25,30,23,20,32,27)Polycyclic presentation:
2^10|a^2=f,b^2=e,c^2=ehij,d^2=j,e^2=j,b^a=bg,c^a=cd,c^b=cei,d^a=dh,d^c=dj,e^c=ej,f^b=fi,f^c=fhj,g^a=gi,g^c=gj,g^d=gj,h^b=hj,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72317617, #72317714, #72317853 |
| maximal normal subgroup of | #72317617, #72317714, #72317853 |
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, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);