Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477922 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17126 |
| 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 | |
819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,9,19,32,17,34,18,33)(4,20,22,10,23,41,29,36)(5,24,26,39,27,40,28,11)(15,42,50,58,44,60,45,30)(16,46,25,31,48,37,49,59)(21,38,53,51,54,35,56,63)(43,64,52,62,47,57,55,61);(1,3,7,19,8,17,6,18)(2,9,14,33,12,34,13,32)(4,15,29,50,23,44,22,45)(5,25,28,48,27,49,26,16)(10,30,41,60,36,58,20,42)(11,37,24,31,39,46,40,59)(21,52,56,47,54,55,53,43)(35,61,63,57,38,62,51,64);(1,4,6,22,8,23,7,29)(2,10,13,20,12,36,14,41)(3,15,18,45,17,44,19,50)(5,21,26,53,27,54,28,56)(9,30,32,42,34,58,33,60)(11,38,40,63,39,35,24,51)(16,47,49,52,48,43,25,55)(31,57,37,61,59,64,46,62);(1,5)(2,11,12,39)(3,16,17,48)(4,21)(6,26)(7,28)(8,27)(9,31)(10,35,36,38)(13,40,14,24)(15,43,44,47)(18,49,19,25)(20,51,41,63)(22,53)(23,54)(29,56)(30,57)(32,37)(33,46)(34,59)(42,61)(45,55,50,52)(58,64)(60,62)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385119);