Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C8×D4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(6,8)(9,10,14,20,15,18,28,35,13,19,23,33,12,16,27,30)(11,21,25,36,26,29,38,39,22,17,31,34,24,32,37,40);(2,4)(10,16,19,18)(11,22)(17,32,21,29)(20,30,33,35)(24,26)(25,31)(34,40,36,39)(37,38);(5,6,7,8)(9,11,23,31,15,26,27,37,13,22,14,25,12,24,28,38)(10,17,33,36,18,32,30,39,19,21,20,34,16,29,35,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72287002, #72287665, #72287668 |
| maximal normal subgroup of | #72287002, #72287665, #72287668 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54554);