Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #362481 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2293 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21270 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,15,6,14,33,43,9,16,32,12,7,13,5,30)(3,17,21,35,22,37,53,42,23,44,19,52,20,10,31,36)(4,24,28,55,26,41,58,38,29,39,57,54,27,11,25,40)(18,46,49,61,50,62,64,63,51,34,47,60,48,45,56,59);(1,3,8,21,6,22,33,53,9,23,32,19,7,20,5,31)(2,10,15,36,14,17,43,35,16,37,12,42,13,44,30,52)(4,18,28,49,26,50,58,64,29,51,57,47,27,48,25,56)(11,34,40,60,24,45,55,59,41,46,38,61,39,62,54,63);(1,4,7,27,9,29,6,26)(2,11,14,24,16,41,13,39)(3,18,20,48,23,51,22,50)(5,25,32,57,33,58,8,28)(10,34,17,45,37,46,44,62)(12,38,30,54,15,40,43,55)(19,47,53,64,21,49,31,56)(35,59,42,61,52,63,36,60);(1,5,6,8,9,33,7,32)(2,12,13,43,16,15,14,30)(3,19,20,53,23,21,22,31)(4,25,26,28,29,58,27,57)(10,35,17,42,37,52,44,36)(11,38,39,55,41,40,24,54)(18,47,48,64,51,49,50,56)(34,59,45,61,46,63,62,60)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,d^b=dh,e^b=ei,g^d=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402488);