Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| 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 | #384 | C23 |
| socle | C22 | C2×#354 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G | |
| LECS | G ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,18,20,24)(12,16,22,15)(13,17)(14,19,23,21);(9,11,13,19)(10,14)(12,21,22,20)(15,18)(16,23)(17,24);(1,3,5,4)(2,6,8,7)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=g,d^2=hi,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=dh,e^b=efh,f^b=fi,e^d=ehi,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56897);