Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357552 |
| 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 | #3508 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,12)(4,10)(5,11)(6,20)(7,9)(8,15)(13,27)(14,21)(16,26)(17,24)(18,23)(19,25)(22,28)(29,32)(30,31)(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);(1,3,6,15,7,16,5,14)(2,8,11,12,9,21,20,26)(4,13,17,28,18,29,19,30)(10,22,25,27,23,31,24,32)(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);(2,9)(8,21)(10,23)(11,20)(12,26)(22,31)(24,25)(27,32)(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)(2,10)(3,13)(5,19)(6,17)(7,18)(8,22)(9,23)(11,25)(12,27)(14,30)(15,28)(16,29)(20,24)(21,31)(26,32)(35,44)(39,52)(43,58)(45,57)(51,63)(53,60)(56,64)(59,62);(1,5,7,6)(2,11,9,20)(3,14,16,15)(4,17,18,19)(8,12,21,26)(10,24,23,25)(13,28,29,30)(22,32,31,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264786);