Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143205 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
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 | |
319 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 319 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) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,6)(3,7,8,5)(10,14)(11,12)(15,16)(17,18)(19,23)(20,24)(21,36)(22,27)(25,33)(26,34)(28,38)(29,46)(30,39)(31,40)(32,41)(35,43)(37,45)(42,48)(44,47);(1,3,4,8)(2,5,6,7)(10,15)(11,12)(14,16)(17,19,22,31)(18,23,27,40)(20,32,35,29)(21,28,34,30)(24,41,43,46)(25,44,42,37)(26,39,36,38)(33,47,48,45);(9,10,13,16)(11,15,12,14)(18,24)(19,28)(20,33)(21,34)(23,37)(25,26)(27,43)(29,32)(30,31)(35,48)(36,42)(38,46)(39,41)(40,44)Polycyclic presentation:
2^9|a^2=d,b^2=di,c^2=hi,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, 51726);