Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,9,7,8,10,12,11)(13,14,17,21)(15,24,22,16)(18,26,28,20)(19,25,27,23);(2,4)(6,10)(7,11)(13,15,18,25,19,26,16,14,17,22,28,23,27,20,24,21);(5,7,9,10,8,11,12,6)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi,c^b=ch116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35941);