Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144122 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #2093 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(9,10,12,19,14,33,13,20,16,22,31,40,15,21,30,39)(11,23,24,18,26,37,29,36,28,38,32,35,27,34,25,17);(1,2)(3,5)(4,7)(6,8)(9,11,14,27,16,28,15,26)(10,17,33,35,22,36,21,18)(12,25,13,32,31,29,30,24)(19,34,20,38,40,37,39,23);(2,5)(3,8)(6,7)(9,11)(10,18)(12,29)(13,24)(14,27)(15,26)(16,28)(17,33)(19,38)(20,37)(21,36)(22,35)(23,40)(25,31)(30,32)(34,39)Polycyclic presentation:
2^9|a^2=d,b^2=deg,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=dgi,e^c=egi,f^c=fh,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53435);