Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | #33666 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
340 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(2,5,7,8)(9,10,12,19,15,22,13,20,16,23,32,34,14,21,30,17)(11,24,25,18,28,35,31,37,29,39,40,38,27,33,26,36);(1,2,6,5,4,7,3,8)(10,17)(12,30)(13,32)(14,15)(18,33)(19,21)(20,23)(22,34)(24,36)(25,26)(27,28)(31,40)(35,38)(37,39);(9,11,16,29)(10,18,23,38)(12,31,32,26)(13,25,30,40)(14,28,15,27)(17,33,20,35)(19,39,34,24)(21,37,22,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=i,d^2=gi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44206);