Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246957 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #392 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,16)(3,17,18,46)(4,9,24,38)(5,26,27,49)(6,10,30,13)(8,15,33,14)(11,44,47,21)(12,22,45,19)(20,55,53,52)(23,34,31,35)(25,37,60,36)(28,63,62,61)(29,40,57,51)(32,43,58,42)(39,59,50,64)(41,56,48,54);(1,3,7,18)(2,9,16,38)(4,22,24,19)(5,25,27,60)(6,29,30,57)(8,21,33,44)(10,39,13,50)(11,45,47,12)(14,36,15,37)(17,49,46,26)(20,32,53,58)(23,28,31,62)(34,43,35,42)(40,61,51,63)(41,52,48,55)(54,64,56,59);(2,10)(3,18)(5,27)(6,30)(9,34)(11,40)(12,48)(13,16)(14,42)(15,43)(17,52)(19,22)(21,44)(23,31)(26,61)(32,58)(35,38)(36,39)(37,50)(41,45)(46,55)(47,51)(49,63)(59,64);(1,4,8,25)(2,11,15,17)(3,19,21,5)(6,31,32,64)(7,24,33,60)(9,12,37,26)(10,40,43,52)(13,51,42,55)(14,46,16,47)(18,22,44,27)(20,56,57,62)(23,58,59,30)(28,53,54,29)(34,48,50,61)(35,41,39,63)(36,49,38,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fi,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154161);