Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4×C8 | C23 |
| commutator | C22 | C42×C8 |
| socle, prime radical | C23 | C2×C4×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,7,10,8,9,11,12);(13,14,15,19,16,20,21,26,17,18,22,24,23,25,27,28);(1,2,3,4)(6,9)(10,12)(14,18)(19,24)(20,25)(26,28)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,h^2=i,c^a=cg,c^b=ci224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33723);