Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #144421 |
| commutator, Frattini, prime radical, cosocle | C2×#535 | C23 |
| LCS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1000 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,7,12)(3,8,18,32)(4,9,22,36)(5,10,24,38)(6,26,13,11)(14,42,46,58)(15,29,17,31)(16,30,48,61)(19,49,52,62)(20,53,54,34)(21,35,41,56)(23,37,47,60)(25,40,27,39)(28,45,59,44)(33,51,63,50)(43,64,55,57);(1,3,13,17)(2,8,26,31)(4,19)(5,23,27,48)(6,15,7,18)(9,33)(10,37,40,61)(11,29,12,32)(14,43,46,55)(16,24,47,25)(20,45)(21,51,41,50)(22,52)(28,57,59,64)(30,38,60,39)(34,58)(35,62,56,49)(36,63)(42,53)(44,54);(1,4)(2,9)(3,14,18,46)(5,20)(6,21)(7,22)(8,28,32,59)(10,34)(11,35)(12,36)(13,41)(15,44,17,45)(16,19,48,52)(23,51,47,50)(24,54)(25,43)(26,56)(27,55)(29,42,31,58)(30,33,61,63)(37,62,60,49)(38,53)(39,57)(40,64)Polycyclic presentation:
2^10|a^2=j,b^2=fghi,c^2=hij,e^2=j,g^2=j,b^a=bf,c^a=cd,c^b=ce,d^b=dg,e^a=eh,e^b=egh,e^d=ej,f^b=fi,f^c=fgh,f^d=fij,f^e=fij,g^a=gi,g^b=gij,g^d=gj,g^e=gj,g^f=gj,h^c=hj,i^b=ij,i^c=ij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72317996, #72318183, #72318948, … |
| maximal normal subgroup of | #72317996, #72318183, #72318948, … |
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.10);;
gap> SetPower(coll, 2, F.6*F.7*F.8*F.9);;
gap> SetPower(coll, 3, F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
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, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*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);