Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4,7,8)(5,6)(9,10,14,18,15,12,13,17)(11,19,22,16,23,24,21,20);(1,3)(2,4)(5,7)(6,8)(9,11,13,22,15,23,14,21)(10,16,17,24,12,20,18,19);(1,4)(2,3)(5,8)(6,7)(9,12)(10,15)(11,20)(13,18)(14,17)(16,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,d^b=di,e^b=eh,f^b=fhi,d^c=dh,e^c=ei,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15371);