Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145372 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,6,8)(3,5,7,4)(10,16)(11,20)(12,15)(13,27)(17,33)(18,28)(19,25)(21,36)(22,29)(23,38)(24,37)(26,30)(31,35)(34,39);(2,3)(4,5)(7,8)(9,10)(11,21)(12,25)(13,28)(14,19)(15,16)(17,23)(18,30)(20,35)(22,34)(24,31)(26,32)(27,40)(29,36)(33,37)(38,39);(1,3,5,8,6,7,4,2)(9,11)(10,17,19,21)(12,24)(13,29)(14,23)(15,22)(16,31,25,34)(18,36,28,33)(20,27)(26,38)(30,37)(32,35,40,39)Polycyclic presentation:
2^9|a^2=gi,c^2=dh,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,d^c=di,e^c=eghi,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52617);