Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1464 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,18,13,19,28,36,16,20,30,37,14,29,12,27)(11,21,25,39,23,35,22,33,26,40,31,38,24,17,32,34);(2,5)(6,8)(9,11)(10,17,20,35)(12,22)(13,23)(14,24)(15,31)(16,26)(18,33,37,34)(19,21,29,40)(25,30)(27,39,36,38)(28,32);(1,3,4,7)(2,6,5,8)(9,12,13,15,16,28,14,30)(10,18,19,36,20,37,29,27)(11,22,23,31,26,32,24,25)(17,33,21,38,35,34,40,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,f^b=fi,e^c=ei,f^c=fgi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42393);