Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
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 | #1786 |
| prime radical | C2×#576 | 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 | |
845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10,13,19,15,20,14,31)(11,21,23,36,24,35,30,17)(12,25,27,37,29,39,28,18)(16,26,32,38,34,40,33,22);(9,11)(10,16)(12,26)(13,23)(14,30)(15,24)(17,18)(19,32)(20,34)(21,25)(22,28)(27,38)(29,40)(31,33)(35,39)(36,37);(9,11)(10,17)(12,22)(13,30)(14,23)(15,24)(16,18)(19,35)(20,36)(21,31)(25,33)(26,28)(27,40)(29,38)(32,39)(34,37)Polycyclic presentation:
2^9|a^2=f,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,g^b=gi,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, 36716);