Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #354 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #382 | C23 |
| socle | C22 | C2×#354 |
| prime radical | C24 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
322 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,3,5,4,6,7,8)(9,10)(11,21,24,31)(12,17)(13,28)(14,19,30,34)(15,20)(16,23,32,40)(18,29,35,27)(22,36,39,33)(25,37,26,38);(9,11,14,25,15,26,30,24)(10,16,19,32,20,33,34,36)(12,22,27,40,13,23,29,39)(17,31,18,21,28,37,35,38);(9,12,15,13)(10,17,20,28)(11,22,26,23)(14,27,30,29)(16,31,33,37)(18,34,35,19)(21,36,38,32)(24,39,25,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,e^b=efhi,e^d=ehi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56806);