Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #115631 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C22×H(5,𝔽2) | C23 |
| prime radical | #62842 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
2921 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 2921 conjugacy classes of subgroups — too many to draw. Only its 151 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)(3,13)(4,9,21,35)(5,10,24,38)(6,25,28,11)(8,22)(14,42)(15,31)(16,41)(17,36)(18,40)(19,33,51,59)(20,52,26,34)(23,39,27,37)(29,45)(30,47)(32,43)(44,62)(46,54)(48,57)(49,61)(50,60,53,58)(55,64)(56,63);(1,3,6,16)(2,8,11,30)(4,18,20,49)(5,22,23,47)(7,17,28,48)(9,32,34,56)(10,36,37,57)(12,31,25,54)(13,39,41,38)(14,26,44,21)(15,27,46,24)(19,43,50,63)(29,52,55,35)(33,42,58,62)(40,60,61,59)(45,53,64,51);(1,4)(2,9)(3,14)(5,19)(6,26,28,20)(7,21)(8,29)(10,33)(11,34,25,52)(12,35)(13,40)(15,45)(16,44,48,49)(17,18)(22,43)(23,53,27,50)(24,51)(30,56,54,55)(31,32)(36,42)(37,58,39,60)(38,59)(41,62,57,61)(46,64,47,63)Polycyclic presentation:
2^10|a^2=ij,b^2=e,c^2=gi,d^2=j,b^a=bd,c^a=cg,c^b=cf,d^b=di,d^c=di,e^a=eij,e^c=eh,e^d=ej,f^a=fj,f^b=fh,f^c=fij,f^d=fj,f^e=fj,g^b=gi,g^e=gj,h^a=hj,h^b=hj,h^c=hj,i^b=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72284674, #72284727, #72284909, … |
| maximal normal subgroup of | #72284674, #72284727, #72284909, … |
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.9*F.10);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
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, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);