Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356733 |
| 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 | #3482 | C22 |
| 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)(35,45)(36,50)(37,42)(38,43)(39,44)(40,48)(41,51)(46,60)(47,56)(49,55)(52,57)(53,58)(54,61)(59,62)(63,64);(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)(33,35,38,49,39,47,37,48)(34,40,42,56,44,55,43,45)(36,46,53,63,52,61,51,62)(41,54,57,64,58,60,50,59);(33,36,39,52)(34,41,44,58)(35,46,47,61)(37,51,38,53)(40,54,55,60)(42,57,43,50)(45,59,56,64)(48,62,49,63);(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)(35,47)(40,55)(45,56)(46,61)(48,49)(54,60)(59,64)(62,63);(4,17)(8,24)(13,16)(18,28)(21,23)(25,32)(26,30)(29,31)(33,37,39,38)(34,42,44,43)(35,48,47,49)(36,51,52,53)(40,56,55,45)(41,57,58,50)(46,62,61,63)(54,64,60,59)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,g^c=gi,e^d=ehi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264013);