Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234422 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,8,7)(9,10,15,12)(11,19,22,24)(13,18,14,17)(16,23,20,21);(2,5)(3,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,19)(18,24);(3,8)(5,7)(9,12,15,10)(11,20,22,16)(13,17,14,18)(19,23,24,21);(2,6)(5,7)(9,13,15,14)(10,17,12,18)(11,21,22,23)(16,19,20,24)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,d^c=di,e^c=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141630);