Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357668 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3511 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,11)(3,12,15,28)(4,9,16,8)(5,18,19,10)(7,20,23,14)(13,27,29,26)(17,24,31,25)(21,30,22,32)(33,34,38,42)(35,39,44,52)(36,46,49,61)(37,50,47,41)(40,54,55,48)(43,56,58,64)(45,60,57,53)(51,62,63,59);(1,3,5,14,6,15,19,20)(2,7,10,12,11,23,18,28)(4,13,17,30,16,29,31,32)(8,21,24,26,9,22,25,27)(33,35)(34,39)(36,43)(37,45)(38,44)(40,51)(41,53)(42,52)(46,56)(47,57)(48,59)(49,58)(50,60)(54,62)(55,63)(61,64);(1,4)(2,8)(3,13)(5,17)(6,16)(7,21)(9,11)(10,24)(12,26)(14,30)(15,29)(18,25)(19,31)(20,32)(22,23)(27,28)(33,36,47,54,38,49,37,48)(34,40,50,61,42,55,41,46)(35,43,57,62,44,58,45,59)(39,51,60,64,52,63,53,56);(1,4)(2,9)(3,13)(5,17)(6,16)(7,22)(8,11)(10,25)(12,27)(14,30)(15,29)(18,24)(19,31)(20,32)(21,23)(26,28)(35,44)(39,52)(43,58)(45,57)(51,63)(53,60)(56,64)(59,62);(4,16)(8,9)(13,29)(17,31)(21,22)(24,25)(26,27)(30,32)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=ghi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^c=eh,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264893);