Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,7,8)(9,10,14,21,13,20,30,17,16,23,12,19,15,22,31,33)(11,24,27,32,28,18,34,36,29,37,25,39,26,38,40,35);(1,2,6,5,4,7,3,8)(10,17)(12,30)(13,15)(14,31)(18,32)(19,22)(20,21)(23,33)(24,36)(25,34)(26,28)(27,40)(35,37)(38,39);(9,11)(10,18)(12,27)(13,28)(14,25)(15,26)(16,29)(17,32)(19,36)(20,37)(21,35)(22,24)(23,38)(30,40)(31,34)(33,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43804);