Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510223 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31396 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24809 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,11,6,13,23,44)(3,14,19,38,20,9,17,40)(4,21,25,5,26,29,10,27)(8,32,31,62,30,12,45,43)(15,50,52,36,51,57,59,48)(16,22,53,41,54,61,55,37)(18,39,33,46,58,49,56,64)(24,47,34,35,63,60,42,28);(1,3)(2,9,13,14)(4,22)(5,16,27,54)(6,20)(7,17)(8,33)(10,41)(11,38,44,40)(12,46,32,64)(15,24)(18,31)(19,23)(21,53,29,55)(25,37)(26,61)(28,36,35,48)(30,56)(34,59)(39,62,49,43)(42,52)(45,58)(47,50,60,57)(51,63);(1,4,23,25,6,26,7,10)(2,5,11,21,13,27,44,29)(3,15,19,59,20,51,17,52)(8,34,45,63,30,42,31,24)(9,36,38,57,14,48,40,50)(12,47,43,28,32,60,62,35)(16,39,53,64,54,49,55,46)(18,41,33,22,58,37,56,61);(1,5)(2,10)(3,16)(4,11)(6,27)(7,29)(8,35)(9,37)(12,34)(13,25)(14,41)(15,39)(17,55)(18,57)(19,53)(20,54)(21,23)(22,40)(24,62)(26,44)(28,30)(31,60)(32,42)(33,48)(36,56)(38,61)(43,63)(45,47)(46,52)(49,51)(50,58)(59,64)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=eh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,e^c=ei,g^c=gh,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417419);