Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15391 |
| 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 | |
918 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)(3,15)(4,10)(5,26,29,40)(6,13)(7,12)(8,14)(9,18)(11,28,41,30)(16,43)(17,47,48,32)(19,33)(20,42)(21,31)(22,52,54,56)(23,37)(24,36)(25,38)(27,39,49,44)(34,46)(35,55,62,53)(45,58,64,61)(50,59)(51,60,57,63);(1,3,6,18,8,20,7,19)(2,9,12,15,14,33,13,42)(4,21,23,46,25,50,24,16)(5,27,28,48,29,49,30,17)(10,34,36,31,38,43,37,59)(11,39,40,47,41,44,26,32)(22,45,53,57,54,64,55,51)(35,61,52,63,62,58,56,60);(1,4,8,25)(2,10,14,38)(3,16,20,46)(5,22,29,54)(6,23,7,24)(9,31,33,59)(11,35,41,62)(12,36,13,37)(15,43,42,34)(17,45,48,64)(18,21,19,50)(26,56,40,52)(27,57,49,51)(28,53,30,55)(32,58,47,61)(39,60,44,63);(1,5,6,30,8,29,7,28)(2,11,12,26,14,41,13,40)(3,17,18,49,20,48,19,27)(4,22,23,55,25,54,24,53)(9,32,15,44,33,47,42,39)(10,35,36,56,38,62,37,52)(16,45,21,51,46,64,50,57)(31,58,43,63,59,61,34,60)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336060);