Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10374 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,19,22,24)(12,16)(13,17)(14,18)(15,21,20,23);(9,10)(11,20,22,15)(12,16)(13,17)(14,18)(19,21,24,23);(1,2,4,6)(3,8,7,5)(9,11)(10,15)(12,21)(13,23)(14,22)(16,19)(17,24)(18,20);(9,12,14,13)(10,16,18,17)(11,21,22,23)(15,19,20,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=i,d^2=h,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,g^b=gh,e^c=eh,f^c=fhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145450);