Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #435481 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15527 |
| 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 | |
909 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)(3,14,18,34)(4,10,22,37)(5,24)(6,12)(7,29)(8,13)(9,17,33,30)(11,28)(15,42)(16,46,47,32)(19,50)(20,52,55,57)(21,36,23,40)(25,43,49,38)(26,39)(27,41)(31,45)(35,54,61,56)(44,58)(48,59)(51,64)(53,62)(60,63);(1,3)(2,9,13,33)(4,19,22,15)(5,25,27,49)(6,17)(7,30)(8,18)(10,31)(11,38)(12,14,29,34)(16,26,47,28)(20,53)(21,48,23,45)(24,46)(32,41)(35,60,61,62)(36,50)(37,59)(39,43)(40,42)(44,54)(51,56)(52,64,57,58)(55,63);(1,4,7,21,8,22,6,23)(2,10,12,40,13,37,29,36)(3,15,30,45,18,19,17,48)(5,20,26,56,27,55,28,54)(9,31,14,42,33,59,34,50)(11,35,41,52,39,61,24,57)(16,44,25,53,47,51,49,63)(32,58,43,60,46,64,38,62);(1,5)(2,11)(3,16)(4,20)(6,28)(7,26)(8,27)(9,32)(10,35)(12,41)(13,39)(14,43)(15,44)(17,49)(18,47)(19,51)(21,56)(22,55)(23,54)(24,29)(25,30)(31,58)(33,46)(34,38)(36,57)(37,61)(40,52)(42,60)(45,53)(48,63)(50,62)(59,64)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342678);