Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6462912 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,14)(4,20)(5,11)(6,30)(7,12)(8,13)(9,17)(10,24)(15,42)(16,38)(18,33)(19,41)(21,49)(22,48)(23,37)(25,36)(26,55)(27,39)(28,56)(29,40)(31,45)(32,47)(34,44)(35,54)(43,59)(46,62)(50,64)(51,57)(52,61)(53,60)(58,63);(1,3,8,19)(2,9,13,33)(4,21,25,44)(5,26,27,47)(6,18,7,17)(10,34,37,49)(11,38,39,62)(12,41,30,14)(15,24,45,23)(16,28,46,29)(20,31,36,42)(22,51,52,43)(32,56,55,40)(35,59,60,57)(48,58,61,64)(50,54,63,53);(1,4,6,24,8,25,7,23)(2,10,12,20,13,37,30,36)(3,15,17,44,19,45,18,21)(5,22,28,54,27,52,29,53)(9,31,14,34,33,42,41,49)(11,35,40,48,39,60,56,61)(16,43,47,63,46,51,26,50)(32,57,38,58,55,59,62,64);(1,5,7,29,8,27,6,28)(2,11,12,40,13,39,30,56)(3,16,18,26,19,46,17,47)(4,22,23,53,25,52,24,54)(9,32,14,38,33,55,41,62)(10,35,20,48,37,60,36,61)(15,43,21,50,45,51,44,63)(31,57,34,58,42,59,49,64);(5,27)(11,39)(16,46)(22,52)(26,47)(28,29)(32,55)(35,60)(38,62)(40,56)(43,51)(48,61)(50,63)(53,54)(57,59)(58,64)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370109);