Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator | C22 | C22×C4×C8 |
| socle | C23 | C23×C8 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,12,16,13,17,20,23,14,11,15,18,21,19,22,24);(2,4)(5,6,8,7);(5,7,8,6)(10,11)(16,18)(17,19)(23,24);(9,11,12,18,13,19,20,24,14,10,15,16,21,17,22,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,h^2=i,b^a=be,c^a=ci,d^c=di200 conjugacy classes — view the full character table.
| maximal subgroup of | #72318476, #72318478, #72318484, … |
| maximal normal subgroup of | #72318476, #72318478, #72318484, … |
| maximal quotient of | #72322577 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400418);