Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8608 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #380 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,7,10,8,9,11,12)(14,17)(15,18)(19,20);(1,2,3,4)(6,9)(7,11);(13,14,18,20,16,19,15,17)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ei,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^b=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16233);