Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503983 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #26232 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #28503 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
560 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)(3,16)(4,22)(5,29)(6,14)(7,13)(8,33)(9,15)(10,19)(11,26)(12,30)(17,49)(18,48)(20,37)(21,39)(23,41)(24,42)(25,45)(27,60)(28,43)(31,46)(32,40)(34,61)(35,44)(36,53)(38,54)(47,51)(50,56)(52,63)(55,62)(57,59)(58,64);(1,3,9,21)(2,10)(4,23)(5,18,31,20)(6,19,7,32)(8,34,30,54)(11,41,45,64)(12,37)(13,16)(14,39)(15,40)(17,24)(22,56,43,62)(25,50)(26,55)(27,53)(28,58)(29,38)(33,48)(35,52)(36,44,49,57)(42,47,60,63)(46,61)(51,59);(1,4,7,26,9,28,6,25)(2,11,14,22,15,45,13,43)(3,17,19,51,21,53,32,52)(5,24,8,35,31,27,30,59)(10,36,16,47,40,49,39,63)(12,42,46,57,33,60,29,44)(18,50,54,58,20,55,34,23)(37,41,38,62,48,64,61,56);(1,5,6,30,9,31,7,8)(2,12,13,29,15,33,14,46)(3,18,19,54,21,20,32,34)(4,24,25,59,28,27,26,35)(10,37,16,38,40,48,39,61)(11,42,43,44,45,60,22,57)(17,50,51,58,53,55,52,23)(36,41,47,62,49,64,63,56)Polycyclic presentation:
2^9|b^2=fh,c^2=f,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411179);