Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356241 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #537 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,15,27)(4,8,17,24)(5,19,12,9)(7,20,22,14)(13,26,16,30)(18,32,28,25)(21,31,23,29)(33,34)(35,43)(36,41)(37,39)(38,50)(40,46)(42,49)(44,57)(45,56)(47,53)(48,61)(51,52)(54,59)(55,60)(58,64)(62,63);(1,3,12,20,6,15,5,14)(2,7,19,27,10,22,9,11)(4,16,28,29,17,13,18,31)(8,23,32,26,24,21,25,30)(34,39)(40,52)(41,53)(42,50)(43,56)(54,63)(55,61)(57,64);(1,4)(2,8)(3,13)(5,18)(6,17)(7,21)(9,25)(10,24)(11,26)(12,28)(14,29)(15,16)(19,32)(20,31)(22,23)(27,30)(33,35,37,45)(34,40,39,52)(36,44,47,58)(38,51,49,46)(41,54,53,63)(42,56,50,43)(48,62,60,59)(55,64,61,57);(4,17)(8,24)(13,16)(18,28)(21,23)(25,32)(26,30)(29,31)(33,36)(34,41)(35,44)(37,47)(38,48)(39,53)(40,54)(42,55)(43,57)(45,58)(46,59)(49,60)(50,61)(51,62)(52,63)(56,64);(36,47)(41,53)(44,58)(48,60)(54,63)(55,61)(57,64)(59,62)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263488);