Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347544 |
| 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 | #62829 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,14,19,33)(4,21,24,52)(5,26,29,11)(6,22,25,10)(8,13,32,41)(9,31,36,17)(15,37,46,35)(16,47,27,42)(18,43,20,44)(23,54,56,51)(28,38,57,53)(30,58,60,40)(34,48,39,59)(45,61,55,62)(49,64,50,63);(1,3)(2,9)(4,15)(5,27)(6,31)(7,19)(8,20)(10,33)(11,39)(12,36)(13,37)(14,22)(16,29)(17,25)(18,32)(21,43)(23,55)(24,46)(26,34)(28,48)(30,49)(35,41)(38,42)(40,61)(44,52)(45,56)(47,53)(50,60)(51,64)(54,63)(57,59)(58,62);(1,2)(3,9)(4,22)(5,11)(6,21)(7,12)(8,13)(10,24)(14,15)(16,34)(17,44)(18,35)(19,36)(20,37)(23,53)(25,52)(26,29)(27,39)(28,51)(30,40)(31,43)(32,41)(33,46)(38,56)(42,45)(47,55)(48,64)(49,61)(50,62)(54,57)(58,60)(59,63);(1,4,7,24)(2,10,12,22)(3,15,19,46)(5,23,29,56)(6,8,25,32)(9,33,36,14)(11,38,26,53)(13,21,41,52)(16,45,27,55)(17,18,31,20)(28,30,57,60)(34,47,39,42)(35,44,37,43)(40,51,58,54)(48,49,59,50)(61,64,62,63);(1,5,8,30)(2,11,13,40)(3,16,20,50)(4,23,25,57)(6,28,24,56)(7,29,32,60)(9,34,37,62)(10,38,21,51)(12,26,41,58)(14,42,44,64)(15,45,17,48)(18,49,19,27)(22,53,52,54)(31,59,46,55)(33,47,43,63)(35,61,36,39)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254771);