Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6359112 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #61089 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,22)(12,18)(13,20)(14,19)(15,21)(16,23)(17,24);(9,11,15,17)(10,16,21,12)(13,18,14,23)(19,24,20,22);(2,5)(3,8)(6,7)(9,12)(10,17)(11,21)(13,24)(14,22)(15,16)(18,19)(20,23);(9,12)(10,18)(11,20)(13,22)(14,24)(15,16)(17,19)(21,23);(9,13,15,14)(10,19,21,20)(11,18,17,23)(12,22,16,24)Polycyclic presentation:
2^9|a^2=hi,b^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266322);