Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,8)(9,10,15,19,14,18,16,21)(11,20,22,12,17,13,24,23);(2,4)(9,11,13,21,16,22,12,10,14,17,20,19,15,24,23,18);(5,6,7,8)(9,12,16,20,14,23,15,13)(10,11,21,24,18,17,19,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319715, #72319737, #72319771, … |
| maximal normal subgroup of | #72319715, #72319737, #72319771, … |
| maximal quotient of | #72322393 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54626);