Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
536 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(9,10,14,12,15,18,29,25)(11,19,22,33,23,34,37,40)(13,26,27,38,28,17,32,24)(16,30,20,35,31,39,36,21);(1,2,3,5,4,6,7,8)(9,11)(10,16)(12,20)(13,21,28,35)(14,22)(15,23)(17,19,26,34)(18,31)(24,33,38,40)(25,36)(27,30,32,39)(29,37);(9,12,14,18,15,25,29,10)(11,20,22,31,23,36,37,16)(13,24,27,26,28,38,32,17)(19,21,33,30,34,35,40,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ci,d^a=di,f^a=fgi,d^b=dg,g^b=gi,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35511);