Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C4×C16 | C23 |
| socle, prime radical | C22 | D4×C16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
128 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,20,25,21,26,32,40,22,19,28,29,34,30,43,44,23,27,33,41,35,42,46,48,36,31,24,37,38,45,39,47);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9);(2,7)(6,15)(8,12)(9,16)(17,19,20,29,21,30,32,44,22,27,28,41,34,42,43,48,23,31,33,37,35,45,46,47,36,18,24,25,38,26,39,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=cfi,d^a=df,d^b=df152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58296);