Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7493 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C2×D4 | C4×C8 |
| Frattini, cosocle | C4×#127 | C22 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C4×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 2 generators on 24 points:
(1,2,4,7,6,8,5,3)(9,10)(11,18,20,14,19,23,21,24)(12,15)(13,16)(17,22);(1,3,5,8,6,7,4,2)(9,11)(10,14,16,24)(12,21)(13,19)(15,23,17,18)(20,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=ghi,f^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,c^b=ci,d^b=dfi,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,f^c=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1343);