Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351101 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| 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 | |
2531 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)(3,12)(4,9)(5,10)(6,19)(7,11)(8,14)(13,27)(15,22)(16,24)(17,25)(18,26)(20,23)(21,29)(28,31)(30,32)(33,34)(35,46)(36,50)(37,42)(38,45)(39,44)(40,43)(41,48)(47,61)(49,56)(51,64)(52,60)(53,59)(54,58)(55,62)(57,63);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,28)(17,30)(18,29)(24,27)(25,31)(26,32)(33,35,40,36)(34,41,43,56)(37,47,52,51)(38,49,39,48)(42,55,60,57)(44,50,45,46)(53,63,54,62)(58,64,59,61);(1,4,7,17)(2,9,11,25)(3,13,15,30)(5,18,6,16)(8,21,23,31)(10,26,19,24)(12,27,22,32)(14,29,20,28)(33,36,40,35)(34,41,43,56)(37,47,52,51)(38,48,39,49)(42,57,60,55)(44,50,45,46)(53,63,54,62)(58,61,59,64);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,23,12)(9,24,25,26)(13,28,30,29)(21,27,31,32)(33,37,40,52)(34,42,43,60)(35,47,36,51)(38,53,39,54)(41,55,56,57)(44,58,45,59)(46,61,50,64)(48,62,49,63);(2,11)(8,23)(9,25)(10,19)(12,22)(21,31)(24,26)(27,32)(34,43)(37,52)(41,56)(44,45)(46,50)(47,51)(53,54)(62,63)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,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, 6258373);