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 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8)(3,13)(4,18)(5,22)(9,28)(10,33)(11,25)(12,23)(14,35)(16,42)(17,45)(19,27)(21,40)(24,41)(26,32)(29,52)(30,50)(31,56)(34,58)(36,38)(37,39)(43,54)(44,61)(46,55)(48,51)(53,60)(57,59)(62,64);(1,2)(3,9)(4,19)(5,23)(6,12)(7,11)(8,22)(10,18)(13,32)(14,29)(15,30)(16,43)(17,46)(20,27)(21,48)(24,50)(25,41)(26,39)(28,45)(31,38)(33,47)(34,40)(35,55)(36,60)(37,52)(42,57)(44,53)(49,51)(54,61)(56,64)(58,63)(59,62);(3,14)(5,24)(8,25)(9,29)(13,35)(16,38)(17,45)(20,49)(22,41)(23,50)(26,52)(27,51)(28,46)(31,43)(32,55)(33,58)(36,42)(37,39)(44,64)(47,63)(53,56)(54,59)(57,60)(61,62);(1,3,15,35)(2,9,30,55)(4,16,40,42)(5,17,41,45)(6,14,7,13)(8,26,50,52)(10,31,48,60)(11,32,12,29)(18,38,21,36)(19,43,34,57)(20,44,63,64)(22,39,24,37)(23,46,25,28)(27,53,58,56)(33,59,51,54)(47,62,49,61);(1,4,7,21)(2,10,11,34)(3,16,13,36)(5,20,24,49)(6,18,15,40)(8,27,25,51)(9,31,32,57)(12,19,30,48)(14,38,35,42)(17,44,37,61)(22,47,41,63)(23,33,50,58)(26,53,28,54)(29,43,55,60)(39,62,45,64)(46,59,52,56)Polycyclic presentation:
2^9|d^2=hi,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, 6254782);