Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #398772 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2439 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15220 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
887 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)(3,13,16,8)(4,18,21,9)(5,23)(6,26)(7,11)(10,24)(12,27)(14,31)(15,39,40,51)(17,30,42,36)(19,28)(20,48,49,52)(22,33,45,44)(25,53)(29,41,35,59)(32,50,43,62)(34,54)(37,60)(38,56)(46,63)(47,58)(55,64)(57,61);(1,3,6,17,7,16,27,42)(2,8,26,36,11,13,12,30)(4,19,45,38,21,14,22,47)(5,15,24,41,25,40,54,59)(9,31,33,58,18,28,44,56)(10,29,53,39,34,35,23,51)(20,46,62,61,49,37,50,64)(32,57,52,60,43,55,48,63);(1,4)(2,9)(3,14)(5,20)(6,22)(7,21)(8,28)(10,32)(11,18)(12,33)(13,31)(15,37)(16,19)(17,38)(23,52)(24,50)(25,49)(26,44)(27,45)(29,55)(30,56)(34,43)(35,57)(36,58)(39,60)(40,46)(41,61)(42,47)(48,53)(51,63)(54,62)(59,64);(1,5,6,24,7,25,27,54)(2,10,12,23,11,34,26,53)(3,15,17,41,16,40,42,59)(4,20,22,50,21,49,45,62)(8,29,30,51,13,35,36,39)(9,32,33,52,18,43,44,48)(14,37,38,61,19,46,47,64)(28,55,56,63,31,57,58,60)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323481);