Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #133925 |
| commutator | C22×#41 | C22×C4 |
| Frattini, cosocle | C2×#535 | C23 |
| prime radical | #35527 | C22 |
| LCS | G ▷ C22×#41 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35527 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
2442 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 2442 conjugacy classes of subgroups — too many to draw. Only its 95 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:
(1,2,6,7)(3,12)(4,16,18,9)(5,11,20,10)(8,22)(13,29)(14,21)(15,27)(17,28)(19,26,32,25)(23,30)(24,31);(2,7)(3,13)(5,20)(8,23)(9,16)(12,28)(14,30)(15,17)(19,32)(21,24)(22,31)(27,29);(1,3)(2,8,10,12,7,21,11,27)(4,17,18,13)(5,14)(6,15)(9,24,25,29,16,23,26,28)(19,30,32,31)(20,22)Polycyclic presentation:
2^10|a^2=gij,c^2=e,e^2=i,h^2=j,b^a=bj,c^a=cf,c^b=cd,d^a=dj,d^c=dg,e^a=eghj,e^b=eg,e^d=ej,f^a=fj,f^c=fgh,f^d=fj,f^e=fi,g^a=gj,g^c=gj,h^b=hj,h^c=hi,h^f=hj,i^a=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72319170, #72319516, #72319838, … |
| maximal normal subgroup of | #72319170, #72319516, #72319838, … |
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.7*F.9*F.10);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
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);