Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143205 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 512 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,12,15)(11,19,21,24)(13,18,14,17)(16,20,23,22)(25,26)(27,30)(28,37)(29,33)(31,34)(32,35)(36,39)(38,40);(1,3,4,8)(2,5,6,7)(10,15)(11,20)(13,14)(16,19)(21,22)(23,24);(9,11,12,21)(10,16,15,23)(13,20,14,22)(17,24,18,19)(26,30)(27,34)(28,35)(31,32)(33,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=d,b^2=dh,c^2=h,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,f^b=fh,e^c=eg,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51748);