Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352701 |
| 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 | C22×C4 | C25 |
| prime radical | #62903 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,26)(3,14,27,52)(4,11,28,36)(5,22,29,35)(6,13,7,10)(9,30,51,16)(12,24,50,25)(15,41,53,63)(17,40,18,39)(19,38,21,34)(20,47,54,59)(23,33,45,31)(32,55,57,43)(37,49,60,46)(42,61,44,62)(48,58,64,56);(1,3,6,18)(2,9,13,33)(4,15,19,42)(5,23,25,16)(7,17,8,27)(10,31,26,51)(11,32,38,58)(12,39,22,14)(20,48,46,43)(21,44,28,53)(24,30,29,45)(34,56,36,57)(35,52,50,40)(37,61,47,41)(49,55,54,64)(59,63,60,62);(5,24)(12,35)(14,40)(16,45)(20,49)(22,50)(23,30)(25,29)(37,59)(39,52)(41,62)(43,64)(46,54)(47,60)(48,55)(61,63);(2,10)(4,19)(9,31)(11,36)(12,35)(13,26)(14,40)(15,42)(20,46)(21,28)(22,50)(32,57)(33,51)(34,38)(37,60)(39,52)(41,63)(43,48)(44,53)(47,59)(49,54)(55,64)(56,58)(61,62);(1,4,7,21)(2,11,10,34)(3,15,17,44)(5,20,24,49)(6,19,8,28)(9,32,31,56)(12,37,35,59)(13,38,26,36)(14,41,40,62)(16,43,45,64)(18,42,27,53)(22,47,50,60)(23,48,30,55)(25,46,29,54)(33,58,51,57)(39,61,52,63)Polycyclic presentation:
2^9|a^2=hi,b^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260093);