Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#12316 |
| commutator, Frattini, cosocle | C4×C16 | C24 |
| LCS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#2170 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,12,31,15,26,13,8,10,6,32,7,14,30,41)(3,16,21,46,56,48,57,34,19,49,22,61,23,47,60,63)(4,24,27,40,18,52,53,43,29,62,25,11,28,42,55,44)(9,33,35,54,38,17,36,51,39,58,45,59,37,50,64,20);(3,17)(4,25)(5,30)(6,26)(7,31)(9,34)(11,24)(12,41)(13,32)(14,15)(16,36)(18,53)(19,50)(20,57)(21,54)(22,59)(23,58)(27,29)(28,55)(33,56)(35,48)(37,61)(38,46)(39,63)(40,62)(42,44)(43,52)(45,47)(49,64)(51,60);(1,3,18,54,30,60,27,33,7,23,4,20,6,22,55,50,8,19,28,59,26,57,25,58,31,56,29,51,5,21,53,17)(2,9,11,16,32,45,52,61,15,38,44,48,41,64,62,63,10,39,40,49,12,35,42,46,14,37,43,47,13,36,24,34);(2,10)(3,19)(11,40)(12,32)(13,41)(14,15)(17,50)(20,51)(21,22)(23,56)(24,62)(33,58)(42,52)(43,44)(54,59)(57,60)Polycyclic presentation:
2^10|a^2=f,c^2=efgj,e^2=gi,f^2=hj,g^2=ij,h^2=j,i^2=j,b^a=bfgj,c^a=ce,c^b=cefhj,d^a=dj,d^c=dj,e^a=eg,e^b=eg,f^b=fh,g^a=gi,g^b=gi,h^b=hj,i^a=ij,i^b=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72225576, #72226897, #72226902 |
| maximal normal subgroup of | #72225576, #72226897, #72226902 |
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.6);;
gap> SetPower(coll, 3, F.5*F.6*F.7*F.10);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);