Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11757 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | C4×#99 |
| prime radical | #392 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
430 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 108 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)(3,5)(4,7)(6,8)(9,10,14,12,15,18,30,25)(11,19,22,35,23,34,37,38)(13,26,28,39,29,17,33,24)(16,31,20,27,32,40,36,21);(2,5)(3,8)(6,7)(9,11)(10,16,18,32)(12,20,25,36)(13,27,29,21)(14,22)(15,23)(17,34)(19,26)(24,38)(28,40,33,31)(30,37)(35,39);(9,12,14,18,15,25,30,10)(11,20,22,32,23,36,37,16)(13,24,28,26,29,39,33,17)(19,21,35,31,34,27,38,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh,h^b=hi,f^c=fi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36055);