Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×C22⋊C8 |
| prime radical | C2×#41 | C42 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
371 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 371 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 3 generators on 40 points:
(1,2,4,3,5,6,8,7)(9,10)(11,20)(12,25)(13,27)(14,17)(15,19)(16,22)(18,29)(21,33)(23,37)(24,30)(26,32)(28,39)(31,38)(34,35)(36,40);(9,11,13,23,14,24,28,38,15,21,29,35,12,22,26,36)(10,16,18,31,17,20,32,34,19,30,27,40,25,33,39,37);(1,3,4,6,5,7,8,2)(11,21)(16,30)(20,33)(22,24)(23,35)(31,40)(34,37)(36,38)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg,d^a=di,f^a=fi,c^b=ci128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35472);