Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356533 |
| 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 | #62129 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1487 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)(3,11,15,25)(4,8,16,22)(5,18,12,9)(7,19,21,14)(13,24,27,32)(17,29,26,23)(20,30,31,28)(33,34,39,44)(35,45,49,55)(36,50,52,57)(37,42,38,43)(40,48,56,47)(41,53,58,51)(46,60,62,54)(59,63,64,61);(1,3,12,19,6,15,5,14)(2,7,18,25,10,21,9,11)(4,13,26,30,16,27,17,28)(8,20,29,32,22,31,23,24)(33,35,37,47,39,49,38,48)(34,40,43,45,44,56,42,55)(36,46,51,61,52,62,53,63)(41,54,50,59,58,60,57,64);(33,36,39,52)(34,41,44,58)(35,46,49,62)(37,51,38,53)(40,54,56,60)(42,57,43,50)(45,59,55,64)(47,61,48,63);(1,4,6,16)(2,8,10,22)(3,13,15,27)(5,17,12,26)(7,20,21,31)(9,23,18,29)(11,24,25,32)(14,28,19,30)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,52,53)(40,55,56,45)(41,57,58,50)(46,61,62,63)(54,64,60,59);(4,16)(8,22)(13,27)(17,26)(20,31)(23,29)(24,32)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,g^c=gi,e^d=ehi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263825);