Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,18)(12,22)(13,16)(14,17)(15,19)(20,24)(21,23);(9,11,13,19,14,21,12,20)(10,15,16,23,17,24,22,18);(1,2,3,4)(11,19)(12,13)(15,18)(16,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,d^c=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31162);