Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343485 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,15,12)(11,19,22,24)(13,17,14,18)(16,23,20,21);(2,5)(3,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19);(9,12,15,10)(11,20,22,16)(13,17,14,18)(19,21,24,23);(9,13,15,14)(10,17,12,18)(11,21,22,23)(16,24,20,19);(2,5)(6,8)(9,13,15,14)(10,18,12,17)(11,21,22,23)(16,19,20,24)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=i,f^2=i,g^2=hi,b^a=bf,e^a=eh,f^a=fh,f^b=fi,g^b=ghi,d^c=di98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251372);