Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1823 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,6,7,4,3,5,8)(9,10,13,20,15,18,14,30)(11,21,24,34,22,16,29,33)(12,25,27,36,28,17,26,19)(23,35,40,37,38,31,39,32);(2,3)(7,8)(9,11)(10,16)(12,23)(13,29)(14,24)(15,22)(17,35)(18,21)(19,32)(20,34)(25,31)(26,40)(27,39)(28,38)(30,33)(36,37);(1,3,4,2)(5,7,6,8)(10,17,18,25)(11,22)(12,26)(13,14)(16,31,21,35)(19,20,36,30)(23,39)(27,28)(32,33,37,34)(38,40)Polycyclic presentation:
2^9|a^2=f,c^2=d,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ch,f^b=fi,e^c=eg,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42482);