Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139906 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | #2309 | C23 |
| prime radical | C2×#535 | C2×C4 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35327 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
912 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 912 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,11)(3,8,15,28)(4,9,20,32,21,33,24,37)(5,10,23,36)(7,12,25,38)(13,26,40,52,41,53,44,56)(14,27,43,55)(16,29,45,57)(17,46,47,62,42,30,51,58)(18,35,48,60)(19,34,49,61,22,31,50,59)(39,54,63,64);(1,3,7,16)(2,8,12,29)(4,17)(5,14,18,39)(6,15,25,45)(9,30)(10,27,35,54)(11,28,38,57)(13,22)(19,41)(20,47)(21,42)(23,43,48,63)(24,51)(26,34)(31,53)(32,58)(33,46)(36,55,60,64)(37,62)(40,50)(44,49)(52,61)(56,59);(1,4,18,19,7,21,5,22)(2,9,10,34,12,33,35,31)(3,13,39,17,16,41,14,42)(6,24,48,50,25,20,23,49)(8,26,27,46,29,53,54,30)(11,37,36,59,38,32,60,61)(15,44,63,51,45,40,43,47)(28,56,55,58,57,52,64,62)Polycyclic presentation:
2^10|a^2=f,b^2=hj,c^2=dhi,d^2=j,f^2=h,b^a=bh,c^a=ce,c^b=cd,d^a=dh,d^c=dj,e^a=eg,e^b=eh,e^c=ehj,e^d=ej,f^c=fg,f^e=fi,g^a=gi,g^f=gj,h^c=hj,i^a=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72285412, #72285958, #72285974 |
| maximal normal subgroup of | #72285412, #72285958, #72285974 |
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.8*F.10);;
gap> SetPower(coll, 3, F.4*F.8*F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);