Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#107 |
| prime radical | C23 | C4×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,6,8,3)(9,10,13,20,15,22,31,35,16,18,30,36,14,21,12,19)(11,23,26,34,27,37,25,38,28,39,29,17,24,33,40,32);(2,6)(3,7)(9,11,14,27,16,28,15,24)(10,17,21,32,18,34,22,38)(12,29,30,40,31,26,13,25)(19,37,36,39,35,33,20,23);(1,3,8,6,4,7,5,2)(10,18)(11,24,28,27)(17,32,34,38)(19,35)(20,36)(21,22)(23,37,39,33)(25,26,40,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ci,c^b=ch,d^b=dg,e^b=eg,g^b=gi,h^b=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36353);