Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448891 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15924 |
| 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 | |
814 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)(3,9,19,33,17,35,18,34)(4,20,23,10,24,39,30,38)(5,25,27,40,29,41,28,11)(15,36,21,48,43,59,44,31)(16,45,47,58,26,55,46,32)(22,37,52,49,54,61,53,62)(42,63,51,60,56,57,50,64);(1,3,7,19,8,17,6,18)(2,9,14,34,12,35,13,33)(4,21,30,43,24,44,23,15)(5,26,28,46,29,16,27,47)(10,36,39,31,38,59,20,48)(11,32,25,55,40,58,41,45)(22,51,53,56,54,50,52,42)(37,60,49,63,61,64,62,57);(1,4)(2,10,12,38)(3,15,17,43)(5,22)(6,23)(7,30)(8,24)(9,31)(11,37,40,61)(13,20,14,39)(16,42,26,56)(18,44,19,21)(25,49,41,62)(27,52)(28,53)(29,54)(32,57)(33,36)(34,59)(35,48)(45,64)(46,50,47,51)(55,60)(58,63);(1,5,6,28,8,29,7,27)(2,11,13,25,12,40,14,41)(3,16,18,47,17,26,19,46)(4,22,23,53,24,54,30,52)(9,32,33,55,35,58,34,45)(10,37,20,49,38,61,39,62)(15,42,44,51,43,56,21,50)(31,57,36,60,48,63,59,64)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=e,d^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356088);