Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139434 |
| commutator | C2×#117 | C22×C4 |
| Frattini, cosocle | #2309 | C23 |
| prime radical | C2×#535 | D4 |
| LCS | G ▷ C2×#117 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35327 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
999 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 999 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)(3,13,16,31)(4,18,21,47)(5,23,25,10)(6,11)(7,12)(8,28)(9,20)(14,39)(15,43)(17,32)(19,49)(22,48,40,61)(24,38)(26,55,44,35)(27,36)(29,57)(30,56,54,45)(33,52)(34,53)(37,50,58,41)(42,64,51,46)(59,63)(60,62);(1,3,6,17,7,16,27,28)(2,8,11,31)(4,19,22,51,21,14,40,42)(5,24,26,56,25,15,44,45)(9,29,33,58)(10,30,35,43)(12,32,36,13)(18,46,48,49)(20,41,52,63,53,50,62,57)(23,54,55,38)(34,59,60,37)(39,47,64,61);(1,4)(2,9,12,34)(3,14)(5,20)(6,22)(7,21)(8,29,32,59)(10,18,23,47)(11,33,36,60)(13,37,31,58)(15,41)(16,19)(17,42)(24,50)(25,53)(26,52)(27,40)(28,51)(30,46,54,64)(35,48,55,61)(38,39,43,49)(44,62)(45,63)(56,57)Polycyclic presentation:
2^10|a^2=e,b^2=fghi,c^2=h,d^2=j,f^2=h,b^a=befh,c^a=cd,c^b=chj,d^a=dh,d^b=dhj,e^b=eg,e^c=ehj,e^d=ej,f^a=fghi,f^b=fi,f^e=fij,g^a=gi,g^b=gij,g^f=gj,h^a=hj,i^b=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72285276, #72285958, #72285991 |
| maximal normal subgroup of | #72285276, #72285958, #72285991 |
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.5);;
gap> SetPower(coll, 2, F.6*F.7*F.8*F.9);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);