Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #331031 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #10218 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,13,12,15,18,14,17)(11,19,22,24,23,21,20,16);(2,5)(3,7)(6,8)(9,11)(10,16)(12,21)(13,20)(14,22)(15,23)(17,19)(18,24);(1,2)(3,8)(4,6)(5,7)(10,12)(11,20)(13,14)(17,18)(19,21)(22,23);(2,6)(5,8)(9,12,14,10,15,17,13,18)(11,21,22,16,23,19,20,24)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,g^b=gi,d^c=dei,e^c=eh,f^c=fh,g^c=gi,g^d=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 238227);