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 | #493 |
| 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 | |
296 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10)(11,22)(12,27)(13,30)(14,32)(15,33)(16,21)(17,24)(18,28)(19,23)(20,31)(25,35)(26,40)(29,37)(34,38)(36,39);(1,2,3,7,4,6,8,5)(9,11,23,38,15,26,13,24,16,12,28,39,31,29,14,25)(10,17,32,40,20,34,18,22,21,35,30,37,33,36,19,27);(9,12,28,38,15,29,14,24,16,11,23,39,31,26,13,25)(10,17,30,37,20,34,19,27,21,35,32,40,33,36,18,22)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=egi,d^2=ghi,e^2=gi,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, 48402);