Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C2×C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(10,13)(11,15)(14,16)(17,18,22,23,21,24,19,20);(1,3,5,4)(2,6,7,8)(10,14)(13,16)(17,19,21,22)(18,23,24,20);(9,10,15,16,12,14,11,13);(1,4,5,3)(2,6,7,8)(17,20,19,24,21,23,22,18)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=fh,d^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120609);