Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×D8 |
| prime radical | C23 | #443 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,12,17,25,15,13,18,9,27,11,29,28,33)(7,14,20,30,22,31,8,24,23,32,19,34,21,35,26,36);(2,4)(5,7,10,20,12,22,25,8,13,23,9,19,11,21,28,26)(6,14,16,30,17,31,15,24,18,32,27,34,29,35,33,36);(5,8,25,21,11,20,10,23,13,26,28,22,12,19,9,7)(6,14,27,34,17,31,33,36,18,32,16,30,29,35,15,24)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=eg,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286771, #72286915, #72286922, … |
| maximal normal subgroup of | #72286771, #72286915, #72286922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47544);