Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| 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 | #62881 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,16)(3,17,23,48)(4,8,15,9)(5,12,6,25)(7,26,35,32)(11,37,40,21)(13,29,24,28)(14,33,44,34)(18,49,45,31)(19,46,20,54)(22,43,36,57)(27,58,30,42)(38,55,51,52)(39,60,50,63)(41,61,59,62)(47,53,64,56);(1,3,9,22)(2,11,4,18)(5,19,29,53)(6,20,28,56)(7,31,34,21)(8,36,10,23)(12,38,24,50)(13,39,25,51)(14,43,26,17)(15,45,16,40)(27,60,62,52)(30,63,61,55)(32,48,44,57)(33,37,35,49)(41,64,58,46)(42,54,59,47);(1,4,9,2)(3,18,22,11)(5,12,29,24)(6,25,28,13)(7,32,34,44)(8,16,10,15)(14,35,26,33)(17,37,43,49)(19,38,53,50)(20,51,56,39)(21,57,31,48)(23,45,36,40)(27,59,62,42)(30,41,61,58)(46,63,64,55)(47,52,54,60);(1,5)(2,12)(3,19)(4,24)(6,10)(7,27)(8,28)(9,29)(11,38)(13,15)(14,41)(16,25)(17,46)(18,50)(20,23)(21,52)(22,53)(26,58)(30,35)(31,60)(32,42)(33,61)(34,62)(36,56)(37,55)(39,45)(40,51)(43,64)(44,59)(47,57)(48,54)(49,63);(1,6,10,5)(2,13,16,24)(3,20,23,19)(4,25,15,12)(7,30,35,27)(8,29,9,28)(11,39,40,50)(14,42,44,58)(17,47,48,64)(18,51,45,38)(21,55,37,52)(22,56,36,53)(26,59,32,41)(31,63,49,60)(33,62,34,61)(43,54,57,46)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259736);