Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7217 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #340 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C2×C82 | C22 |
| socle | C22 | #1279 |
| prime radical | C23 | Q8⋊C8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
164 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,5,9,6,10,15,12,7,11,16,3,4,8,14,13)(17,18,21,26,23,29,35,28)(19,30,32,44,34,45,46,39)(20,27,22,25,36,42,37,43)(24,33,38,47,41,48,40,31);(1,3,5,10,6,13,15,11,7,9,16,8,4,12,14,2)(17,19,22,33,21,32,36,47,23,34,37,48,35,46,20,31)(18,24,27,39,28,40,43,45,29,41,42,44,26,38,25,30)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=fgh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132);