Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D4 | SD16 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 608 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(2,3)(4,5)(7,8)(9,10,13,18,15,20,28,17,14,19,29,36,27,35,12,26)(11,21,23,32,25,34,39,38,24,37,40,16,30,31,22,33);(1,2,6,8)(3,4,7,5)(9,11,14,24)(10,16,19,32)(12,23,28,40)(13,22,29,39)(15,30,27,25)(17,34,26,31)(18,37,36,21)(20,38,35,33);(1,3)(2,4)(5,8)(6,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=gi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=egi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309446, #72309457, #72309463, … |
| maximal normal subgroup of | #72309446, #72309457, #72309463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39129);