Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1384 conjugacy classes of subgroups; only the 594 normal subgroups are shown.
This group has 1384 conjugacy classes of subgroups — too many to draw. Only its 594 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) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,29,31)(27,35,36,40)(28,34,30,33)(32,38,39,37);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(26,31)(27,36)(33,34)(37,38);(11,17)(14,20)(18,23)(21,24)(25,27,30,37,29,36,28,38)(26,32,34,35,31,39,33,40);(10,15)(14,20)(16,22)(21,24)(25,28,29,30)(26,33,31,34)(27,37,36,38)(32,35,39,40);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(27,36)(32,39)(35,40)(37,38)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344489);