Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle, prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
760 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)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,19,21,20,22,23,24);(5,6,8,7)(9,10,11,14)(12,13,16,15);(2,4)(5,7,8,6)(10,13)(12,16)(14,15)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,e^c=eh,f^c=fh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27203);