Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C4×D16 |
| prime radical | C2×D16 | C2×C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,5,3,4,6,8,7)(9,10)(11,15)(12,24)(13,27)(14,16)(17,20)(18,21)(19,25)(22,34)(23,28)(26,40)(29,36)(30,35)(31,37)(32,38)(33,39);(2,6)(3,7)(10,15)(11,20)(12,21)(13,25)(16,28)(17,18)(19,34)(22,39)(23,37)(24,29)(26,38)(27,30)(31,32)(33,40)(35,36);(1,3,5,6,4,7,8,2)(10,16)(15,28)(17,31)(18,32)(19,27)(24,40)(29,33)(30,34)Polycyclic presentation:
2^9|a^2=e,c^2=e,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=cgi,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=df,f^b=fh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58320);