Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35250 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4190 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)(3,4)(5,6)(7,8)(12,20)(15,21)(17,24)(19,22);(9,10,13,16)(11,17,18,24)(12,21,20,15)(14,22,23,19);(2,4)(9,11)(10,14)(12,22)(13,18)(15,24)(16,23)(17,21)(19,20);(6,8)Polycyclic presentation:
2^9|b^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301267);