Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458125 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2537 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
718 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 718 conjugacy classes of subgroups — too many to draw. Only its 202 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:
(1,2,4,5)(3,6,7,8)(9,10,14,19)(11,22,24,17)(12,26,29,38)(13,31,25,34)(15,21,16,20)(18,32,37,30)(23,35,39,40)(27,36,28,33);(2,5)(6,8)(9,11,15,25,14,24,16,13)(10,17,21,31,19,22,20,34)(12,27,30,39,29,28,32,23)(18,35,38,36,37,40,26,33);(1,3,4,7)(2,6,5,8)(9,12,14,29)(10,18,19,37)(11,23,24,39)(13,27,25,28)(15,32,16,30)(17,33,22,36)(20,38,21,26)(31,40,34,35);(3,7)(6,8)(9,13,16,24,14,25,15,11)(10,17,21,31,19,22,20,34)(12,28,30,23,29,27,32,39)(18,36,26,35,37,33,38,40)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383049);