Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #505192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #27243 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3584 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1765 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
629 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)(4,21)(5,11)(6,12)(7,29)(8,14)(9,17)(10,24)(13,32)(16,46)(18,51)(19,36)(20,34)(22,37)(23,38)(25,58)(26,39)(27,42)(28,41)(30,35)(31,40)(33,49)(43,53)(44,59)(45,48)(47,61)(50,63)(52,62)(54,55)(56,64)(57,60);(1,3)(2,9,14,36)(4,22,26,56)(5,17)(6,19)(7,30)(8,20)(10,37)(11,34,12,15)(13,43,29,51)(16,25,49,59)(18,27)(21,54)(23,55,24,52)(28,53)(31,47,57,48)(32,50)(33,60)(35,42,63,41)(38,64)(39,62)(40,46)(44,45)(58,61);(1,4,5,23,8,26,6,24)(2,10,12,39,14,38,11,21)(3,16,17,48,20,49,19,47)(7,31,27,59,32,57,28,25)(9,33,15,45,36,46,34,61)(13,44,42,40,29,58,41,60)(18,52,50,22,53,55,30,56)(35,62,43,64,63,54,51,37);(2,11,14,12)(4,23,26,24)(7,32)(9,34,36,15)(10,38)(13,42,29,41)(16,47,49,48)(18,53)(21,39)(22,55,56,52)(25,57,59,31)(27,28)(30,50)(35,43,63,51)(37,64)(54,62)Polycyclic presentation:
2^9|b^2=fh,c^2=e,d^2=fh,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=ghi,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, 412388);