Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C2×C8×Q8 | C22 |
| socle | C22 | #1282 |
| prime radical | C23 | #336 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8×Q8 ▷ C22×C4 ▷ C22 ▷ C1 | |
198 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,8,5,9,13,16)(3,7,10,15,11,6,12,14)(17,18,20,25,22,19,24,28,23,27,21,26,31,30,32,29);(2,6,9,7)(8,15,16,14)(17,19,21,29,22,27,32,25,23,30,20,28,31,18,24,26)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,d^2=g,e^2=gh,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,d^b=dfh,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 194);