Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C42 | C23 |
| commutator | C23 | C43 |
| socle, prime radical | C24 | C2×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10,12,11,13,14,16,15);(2,4)(17,18,19,22,20,21,23,24);(9,11,12,14,13,15,16,10)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ch,c^b=ci176 conjugacy classes — view the full character table.
| maximal subgroup of | #72321132, #72321162, #72321222, … |
| maximal normal subgroup of | #72321132, #72321162, #72321222, … |
| maximal quotient of | #72322457 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12537);