Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #406022 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14922 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,12,7,14)(3,15,18,38,19,9,29,33)(4,20,23,35,25,36,24,10)(5,26,27,37,28,39,30,11)(16,42,44,31,21,51,49,40)(17,45,46,41,47,56,48,32)(22,34,53,59,54,58,55,50)(43,57,62,61,52,60,63,64);(1,3,7,18,8,19,6,29)(2,9,14,33,12,15,13,38)(4,21,24,49,25,16,23,44)(5,17,27,48,28,47,30,46)(10,31,36,51,35,40,20,42)(11,32,26,56,37,41,39,45)(22,52,53,62,54,43,55,63)(34,57,59,64,58,60,50,61);(1,4,8,25)(2,10,12,35)(3,16,19,21)(5,22,28,54)(6,24,7,23)(9,31,15,40)(11,34,37,58)(13,36,14,20)(17,43,47,52)(18,49,29,44)(26,50,39,59)(27,55,30,53)(32,57,41,60)(33,42,38,51)(45,64,56,61)(46,63,48,62);(1,5)(2,11,12,37)(3,17)(4,22)(6,30)(7,27)(8,28)(9,32,15,41)(10,34,35,58)(13,39,14,26)(16,43)(18,48)(19,47)(20,50,36,59)(21,52)(23,55)(24,53)(25,54)(29,46)(31,57,40,60)(33,56,38,45)(42,61,51,64)(44,63)(49,62)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313219);