Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
3372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4)(9,10)(11,15,16,13)(12,14);(5,6,7,8)(10,13)(14,15);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=bd,c^a=ce,d^a=dh,h^a=hi,f^b=fh,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219252, #72219294, #72235138, … |
| maximal normal subgroup of | #72219252, #72219294, #72235138, … |
| maximal quotient of | #72219251, #72226349, #72235865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53653);