Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7157007 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #44936 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61107 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,14,9,16,8,15)(3,17,21,39,23,50,22,10)(4,24,28,11,30,43,29,44)(5,12,33,46,34,47,20,38)(6,35,19,13,31,48,36,49)(18,37,25,57,53,62,55,40)(26,45,32,41,52,63,58,64)(27,42,59,60,51,61,54,56);(1,3)(2,10)(4,25)(5,31)(6,34)(7,22)(8,21)(9,23)(11,40)(12,13)(14,50)(15,17)(16,39)(18,28)(19,33)(20,36)(24,37)(26,27)(29,53)(30,55)(32,54)(35,46)(38,48)(41,61)(42,64)(43,62)(44,57)(45,56)(47,49)(51,52)(58,59)(60,63);(1,4)(2,11)(3,18,23,53)(5,32,34,58)(6,27)(7,29)(8,28)(9,30)(10,37,39,62)(12,45,47,63)(13,42)(14,24)(15,43)(16,44)(17,40,50,57)(19,54)(20,52,33,26)(21,55,22,25)(31,51)(35,60)(36,59)(38,41,46,64)(48,56)(49,61);(1,5,7,33,9,34,8,20)(2,12,14,46,16,47,15,38)(3,19,21,31,23,36,22,6)(4,26,28,32,30,52,29,58)(10,35,17,13,39,48,50,49)(11,41,43,63,44,64,24,45)(18,51,25,54,53,27,55,59)(37,61,57,56,62,42,40,60);(1,6,8,36,9,31,7,19)(2,13,14,48,16,49,15,35)(3,20,22,34,23,33,21,5)(4,27,28,59,30,51,29,54)(10,38,17,12,39,46,50,47)(11,42,24,56,44,61,43,60)(18,52,25,58,53,26,55,32)(37,41,40,45,62,64,57,63)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ehi,h^a=hi,c^b=ch,d^b=df,e^b=eh,f^b=fi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7064203);