Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356405 |
| 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 | #60993 | C2 |
| 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)(3,11,14,25)(4,8,16,22)(5,18,12,9)(7,19,21,15)(13,24,27,32)(17,29,26,23)(20,30,31,28)(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);(1,3,12,19,6,14,5,15)(2,7,18,25,10,21,9,11)(4,13,26,30,16,27,17,28)(8,20,29,32,22,31,23,24)(34,39)(40,52)(41,53)(42,50)(43,56)(54,63)(55,61)(57,64);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,23)(10,22)(11,24)(12,26)(14,27)(15,28)(18,29)(19,30)(21,31)(25,32)(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);(4,16)(8,22)(13,27)(17,26)(20,31)(23,29)(24,32)(28,30)(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);(3,14)(7,21)(11,25)(13,27)(15,19)(20,31)(24,32)(28,30)(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,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263655);