Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2366 |
| prime radical | C2×#496 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10,15,20,16,21,14,19)(11,22,23,17,24,13,18,12);(1,3,4,8,5,2,6,7)(9,11)(10,17)(12,21)(13,19)(14,23)(15,18)(16,24)(20,22);(9,12,14,13,16,17,15,22)(10,18,20,11,21,23,19,24);(9,13,14,17,16,22,15,12)(10,18,19,24,21,23,20,11)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138666);