Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #446240 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15836 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
806 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)(3,15,19,34)(4,20)(5,26,29,41)(6,12)(7,13)(8,14)(9,18,35,31)(10,23)(11,28,42,30)(16,32,46,58)(17,47)(21,36,45,57)(22,37,54,61)(24,39)(25,38)(27,43)(33,49)(40,48)(44,62)(50,53,60,52)(51,56)(55,64)(59,63);(1,3,6,18,8,19,7,31)(2,9,13,34,14,35,12,15)(4,21,23,46,25,45,24,16)(5,27,28,49,29,48,30,17)(10,36,20,32,39,57,38,58)(11,40,41,33,42,43,26,47)(22,44,52,55,54,63,53,51)(37,56,60,59,61,64,50,62);(1,4,6,23,8,25,7,24)(2,10,13,20,14,39,12,38)(3,16,31,45,19,46,18,21)(5,22,30,53,29,54,28,52)(9,32,15,36,35,58,34,57)(11,37,26,50,42,61,41,60)(17,44,27,55,49,63,48,51)(33,56,43,59,47,64,40,62);(1,5,8,29)(2,11)(3,17,19,49)(4,22,25,54)(6,30,7,28)(9,33)(10,37)(12,41)(13,26)(14,42)(15,43)(16,44,46,63)(18,48,31,27)(20,50)(21,51,45,55)(23,53,24,52)(32,56)(34,40)(35,47)(36,59)(38,60)(39,61)(57,62)(58,64)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353436);