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 | |
273 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,3,7,5,8,4,6)(11,20,22,24)(13,26)(16,31,32,33)(18,34)(21,37,38,35)(23,28,39,36)(25,29)(27,40);(2,6)(3,4)(7,8)(9,10,12,17,15,19,14,30)(11,21,24,37,22,38,20,35)(13,27,25,34,26,40,29,18)(16,28,33,39,32,36,31,23);(2,7)(3,4)(6,8)(9,11)(10,16)(12,24)(13,28)(14,20)(15,22)(17,33)(18,35)(19,32)(21,27)(23,29)(25,39)(26,36)(30,31)(34,37)(38,40)Polycyclic presentation:
2^9|a^2=d,b^2=de,d^2=g,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53414);