Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C24 | C2×C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,7,9,8,10,11,12)(13,14)(15,21,22,18)(16,20)(17,19)(23,27,28,25)(24,26);(13,15,16,23,17,22,24,28)(14,18,20,25,19,21,26,27);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15998);