Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12873 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1823 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
395 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 395 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10,13,20,15,18,14,30)(11,21,23,34,24,16,29,33)(12,25,27,35,28,17,26,19)(22,31,38,37,39,36,40,32);(9,11,15,24)(10,16,18,21)(12,22,28,39)(13,29,14,23)(17,31,25,36)(19,32,35,37)(20,34,30,33)(26,38,27,40);(10,17,18,25)(12,26)(13,14)(16,31,21,36)(19,20,35,30)(22,38)(23,29)(27,28)(32,34,37,33)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,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,f^b=fi,e^c=eg,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237238, #72239683, #72239685, … |
| maximal normal subgroup of | #72237238, #72239683, #72239685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42454);