Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #352122 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
716 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 170 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 — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,15)(13,22)(14,18)(16,20)(17,23)(21,24)(27,33)(28,34)(30,36)(32,37)(35,39)(38,40);(2,5)(6,8)(9,11,13,12,14,20,23,21)(10,15,17,19,18,24,22,16)(25,26,29,31)(27,30,35,38)(28,32,34,37)(33,36,39,40);(11,20)(12,21)(15,24)(16,19)(25,27,34,39,29,35,28,33)(26,30,37,40,31,38,32,36);(1,3,4,7)(2,6,5,8)(9,12,13,20,14,21,23,11)(10,16,17,15,18,19,22,24)(26,31)(30,38)(32,37)(36,40)Polycyclic presentation:
2^9|b^2=gi,c^2=fi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=dhi,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260025);