Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508667 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
462 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 124 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:
(1,2,5,4)(3,7,8,6)(9,10,13,15,16,20,14,19)(11,21,23,33,25,31,24,17)(12,26,28,18,30,34,29,35)(22,32,37,27,39,40,38,36);(1,3,5,8)(2,6,4,7)(9,11)(10,17)(12,27)(13,24)(14,23)(15,31)(16,25)(18,22)(19,21)(20,33)(26,37)(28,32)(29,40)(30,36)(34,38)(35,39);(2,4)(6,7)(9,12,16,30)(10,18,20,35)(11,22,25,39)(13,29,14,28)(15,26,19,34)(17,32,33,40)(21,36,31,27)(23,38,24,37);(1,4,5,2)(3,7,8,6)(9,10,13,15,16,20,14,19)(11,17,24,31,25,33,23,21)(12,26,28,18,30,34,29,35)(22,36,38,40,39,27,37,32)Polycyclic presentation:
2^9|a^2=e,b^2=eh,c^2=i,d^2=fi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gh,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415909);