Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1380 |
| prime radical | C23 | #358 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2,5,7,6,3,4,8)(9,10,13,20,16,23,17,43)(11,24,26,55,29,49,30,65)(12,31,34,67,35,21,36,52)(14,19,37,50,39,51,40,68)(15,41,38,53,42,69,33,22)(18,44,45,32,48,58,70,25)(27,61,59,47,57,54,62,71)(28,63,60,66,64,72,56,46);(1,3,6,2)(4,7,5,8)(9,11,14,27,15,28,34,58,13,26,37,59,38,60,35,25,16,29,39,57,42,64,36,44,17,30,40,62,33,56,12,32)(10,18,21,46,22,47,50,65,43,70,67,72,69,61,19,49,23,48,31,66,53,71,68,55,20,45,52,63,41,54,51,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fgi,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,h^a=hi,c^b=cg98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1606);