Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #140051 |
| commutator | #379 | C22×C4 |
| Frattini, cosocle | #2308 | C23 |
| prime radical | #46996 | C22 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
927 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 927 conjugacy classes of subgroups — too many to draw. Only its 67 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)(3,6)(4,8)(5,9,20,27)(7,10)(11,16)(12,22,18,19)(13,23)(14,31,30,28)(15,24,32,26)(17,25)(21,29);(1,3,6,2)(4,14,16,26,17,30,21,24)(5,18,22,9)(7,13,23,10)(8,15,11,28,25,32,29,31)(12,19,27,20);(1,4,7,17)(2,8)(3,11)(5,15,20,32)(6,21,23,16)(9,26)(10,25)(12,30)(13,29)(14,18)(19,31,22,28)(24,27)Polycyclic presentation:
2^10|a^2=f,b^2=ef,c^2=h,d^2=h,e^2=hi,b^a=be,c^a=cgi,c^b=cd,d^a=dg,d^b=dgh,d^c=dhj,e^b=ehi,e^c=eg,e^d=ei,f^b=fhi,f^d=fj,g^a=gj,g^b=gi,g^d=gj,g^e=gj,h^a=hj,h^b=hj,i^a=ij,i^b=ij,i^c=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72237845, #72237880, #72237904, … |
| maximal normal subgroup of | #72237845, #72237880, #72237904, … |
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*F.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);