Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C42 | C23 |
| commutator | C23 | C2×C4×C8 |
| socle | C24 | C2×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(17,18,21,24,20,23,22,19);(2,4)(9,10,11,13,12,14,15,16)(17,19,22,23,20,24,21,18);(6,8)(18,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,c^b=ch176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12677);