Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #121 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,7)(9,10)(11,13)(12,15)(14,16)(17,18,21,22,20,23,19,24);(5,6,8,7)(9,10)(11,13)(12,15)(14,16)(17,19,20,21)(18,22,23,24);(2,4)(5,7,8,6)(10,13)(11,16)(14,15)(18,22)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dhi,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,e^c=ei,f^c=fh,g^c=ghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27335);