Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395187 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1178 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1178 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(3,7)(5,8)(9,10,15,18,14,20,13,19)(11,21,30,22,25,32,24,16)(12,26,27,34,29,33,28,17)(23,31,37,40,39,36,38,35);(1,2,4,6)(3,8,7,5)(9,11,13,24,14,25,15,30)(10,16,18,21,20,22,19,32)(12,23,27,37,29,39,28,38)(17,31,33,35,34,36,26,40);(1,3,4,7)(2,5,6,8)(9,10,14,20)(11,22,25,16)(12,17,29,34)(13,18,15,19)(21,24,32,30)(23,36,39,31)(26,27,33,28)(35,37,40,38);(9,12,13,27,14,29,15,28)(10,17,18,33,20,34,19,26)(11,23,24,37,25,39,30,38)(16,31,21,35,22,36,32,40)Polycyclic presentation:
2^9|a^2=fg,b^2=egi,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311019);