Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62928 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,29)(3,14,17,41)(4,20,30,37)(5,24,31,26)(6,12,7,13)(9,32,34,16)(10,21,50,23)(11,28,57,27)(15,43,46,62)(18,39,19,42)(22,53,58,38)(25,35,48,36)(33,54,61,64)(40,44,63,47)(45,49,59,60)(51,55,56,52);(1,3,6,18)(2,9,12,35)(4,15,21,44)(5,25,27,16)(7,19,8,17)(10,33,37,60)(11,39,24,14)(13,36,29,34)(20,49,50,61)(22,54,52,45)(23,47,30,46)(26,41,57,42)(28,32,31,48)(38,62,56,40)(43,51,63,53)(55,59,58,64);(1,2,7,13)(3,9,19,36)(4,10,23,20)(5,26,28,11)(6,12,8,29)(14,16,42,48)(15,33,47,49)(17,34,18,35)(21,37,30,50)(22,51,55,38)(24,27,57,31)(25,41,32,39)(40,45,43,64)(44,60,46,61)(52,53,58,56)(54,63,59,62);(1,4,7,23)(2,10,13,20)(3,15,19,47)(5,22,28,55)(6,21,8,30)(9,33,36,49)(11,38,26,51)(12,37,29,50)(14,40,42,43)(16,45,48,64)(17,46,18,44)(24,56,57,53)(25,54,32,59)(27,52,31,58)(34,61,35,60)(39,62,41,63);(4,21)(10,37)(15,44)(20,50)(22,52)(23,30)(33,60)(38,56)(40,62)(43,63)(45,54)(46,47)(49,61)(51,53)(55,58)(59,64)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260120);