Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #253387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #392 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,9,3)(4,21,27,56)(5,29,33,58)(6,12,15,18)(7,36,39,37)(8,40,42,51)(10,43,16,25)(11,49,17,31)(13,52,19,34)(14,35,20,32)(22,60,47,30)(23,62,63,61)(24,59,28,53)(26,45,41,55)(38,48,64,46)(44,50,54,57);(1,3,15,12)(2,6,18,9)(4,22,28,62)(5,11,49,58)(7,37,52,13)(8,20,32,51)(10,44,48,26)(14,35,40,42)(16,54,46,41)(17,31,29,33)(19,39,36,34)(21,23,53,30)(24,61,27,47)(25,45,38,50)(43,55,64,57)(56,63,59,60);(1,4,6,28)(2,10,12,48)(3,16,18,46)(5,30,31,23)(7,38,34,25)(8,41,35,54)(9,27,15,24)(11,50,29,45)(13,53,36,21)(14,22,51,62)(17,57,58,55)(19,59,37,56)(20,47,40,61)(26,32,44,42)(33,60,49,63)(39,64,52,43);(1,5,6,31)(2,11,12,29)(3,17,18,58)(4,23,28,30)(7,35,34,8)(9,33,15,49)(10,45,48,50)(13,20,36,40)(14,37,51,19)(16,55,46,57)(21,47,53,61)(22,59,62,56)(24,60,27,63)(25,54,38,41)(26,43,44,64)(32,52,42,39)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=i,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=eh,f^c=fi,g^c=gh,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 160605);