Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,20,16,21,28,40,15,22,32,33,13,19,23,39)(11,17,12,27,26,35,29,37,25,36,30,38,24,34,31,18);(2,5)(6,8)(9,11,16,26,15,25,13,24)(10,17,21,35,22,36,19,34)(12,28,29,32,30,23,31,14)(18,20,27,40,37,33,38,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,23)(13,31)(14,26)(15,30)(16,29)(17,33)(19,38)(20,36)(21,27)(22,37)(24,32)(25,28)(34,40)(35,39)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=d,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,e^c=ei,f^c=fgi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42410);