Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1970 |
| prime radical | C23 | #495 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
281 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 281 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,15,21,16,23,33,22)(11,24,27,40,28,18,38,39)(12,29,26,37,31,36,25,17)(13,32,30,19,14,34,35,20);(1,2,3,7,4,6,8,5)(9,11,14,26,33,38,30,12,16,28,13,25,15,27,35,31)(10,17,19,24,21,29,34,40,23,37,20,18,22,36,32,39);(9,12,30,27,15,26,14,28,16,31,35,38,33,25,13,11)(10,18,32,37,22,40,20,29,23,24,34,17,21,39,19,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=egi,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,f^b=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48407);