Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #10336 |
| commutator | C23×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C23×#41 | C23 |
| socle | C2 | #121408 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
2001 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 2001 conjugacy classes of subgroups — too many to draw. Only its 163 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:
(1,2,5,10,7,12,22,35)(3,8,14,27,16,29,42,53)(4,9,18,31,20,33,46,57)(6,11,21,34,25,38,49,60)(13,26,39,50,40,51,61,63)(15,36,41,58,23,28,47,52)(17,30,44,55,43,54,62,64)(19,37,45,59,24,32,48,56);(1,3)(2,8)(4,17)(5,14)(6,23,25,15)(7,16)(9,30)(10,27)(11,36,38,28)(12,29)(13,19,40,24)(18,44)(20,43)(21,47,49,41)(22,42)(26,32,51,37)(31,55)(33,54)(34,58,60,52)(35,53)(39,45,61,48)(46,62)(50,56,63,59)(57,64);(1,4)(2,9)(3,13)(5,18)(6,24,25,19)(7,20)(8,26)(10,31)(11,37,38,32)(12,33)(14,39)(15,43,23,17)(16,40)(21,48,49,45)(22,46)(27,50)(28,54,36,30)(29,51)(34,59,60,56)(35,57)(41,62,47,44)(42,61)(52,64,58,55)(53,63)Polycyclic presentation:
2^10|a^2=d,b^2=f,c^2=g,d^2=j,b^a=bf,c^a=cg,c^b=ce,e^a=ei,e^b=eh,e^c=ehi,f^c=fh,f^e=fj,g^b=ghi,g^e=gj,h^b=hj,i^b=ij,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72342843, #72342849, #72342859 |
| maximal normal subgroup of | #72342843, #72342849, #72342859 |
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.4);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, 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);