Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504480 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #26633 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3067 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,34,50,7,16,33,52,9,15,35,51)(3,17,20,39,21,56,59,38,23,55,60,40,22,47,30,10)(4,11,24,42,36,37,62,61,19,44,31,48,25,43,58,64)(5,26,27,12,18,45,32,53,29,49,63,46,28,54,57,41);(1,3,7,23)(2,10,16,38)(4,5,19,29)(6,30,33,59)(8,22,9,21)(11,41,44,53)(12,37,46,43)(13,47,52,56)(14,40,15,39)(17,50,55,51)(18,36,28,25)(20,34,60,35)(24,57,31,32)(26,61,49,64)(27,62,63,58)(42,54,48,45);(1,4)(2,11)(3,18)(5,22)(6,31)(7,19)(8,36)(9,25)(10,12)(13,48)(14,37)(15,43)(16,44)(17,54)(20,57)(21,29)(23,28)(24,33)(26,56)(27,59)(30,63)(32,60)(34,58)(35,62)(38,46)(39,53)(40,41)(42,52)(45,55)(47,49)(50,64)(51,61);(1,5,9,18,7,29,8,28)(2,12,14,41,16,46,15,53)(3,19,21,36,23,4,22,25)(6,32,35,63,33,57,34,27)(10,37,40,44,38,43,39,11)(13,49,50,45,52,26,51,54)(17,48,47,64,55,42,56,61)(20,58,59,31,60,62,30,24)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=fhi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ehi,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411679);