Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #10846 |
| commutator | C23×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C23×#41 | C23 |
| socle | C2 | #123862 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#568 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C22 ▷ C2 ▷ C1 | |
3771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,14)(4,19,22,35)(5,10)(6,12)(7,13)(8,17)(9,21,34,53)(11,20)(15,45,47,64)(16,43)(18,39)(23,52,27,59)(24,44)(25,37)(26,32)(28,41)(29,60,33,63)(30,49)(31,57)(36,56,40,58)(38,54)(42,46,62,55)(48,61,50,51);(1,3)(2,8)(4,15)(5,24)(6,26,28,17)(7,18)(9,33)(10,31)(11,30)(12,39,41,14)(13,32)(16,20)(19,51)(21,55)(22,47)(23,50,27,48)(25,49,54,57)(29,34)(35,61)(36,62,40,42)(37,43,38,44)(45,59,64,52)(46,53)(56,63,58,60);(1,4,20,53,7,22,5,21)(2,9,10,19,13,34,11,35)(3,15,16,46,18,47,24,55)(6,27,54,56,28,23,25,58)(8,29,31,61,32,33,30,51)(12,40,37,59,41,36,38,52)(14,42,44,64,39,62,43,45)(17,50,49,60,26,48,57,63)Polycyclic presentation:
2^10|a^2=e,b^2=h,c^2=fij,d^2=j,f^2=j,b^a=beg,c^a=cd,c^b=cdfj,d^b=dj,e^c=ej,f^a=fj,f^b=fj,g^b=gi,g^c=gj,h^a=hi,h^g=hj,i^b=ij106 conjugacy classes — view the full character table.
| maximal subgroup of | #72312278, #72312714, #72313201, … |
| maximal normal subgroup of | #72312278, #72312714, #72313201, … |
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.8);;
gap> SetPower(coll, 3, F.6*F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);