Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,15,19,14,29)(11,20,22,36,23,37,30,34)(12,24,26,35,28,39,27,17)(16,25,31,38,33,40,32,21);(1,3)(2,5,6,7)(4,8)(9,11)(10,16)(12,25)(13,22)(14,30)(15,23)(17,34)(18,31)(19,33)(20,24)(21,27)(26,38)(28,40)(29,32)(35,36)(37,39);(3,8)(5,7)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,c^b=cghi,d^b=dh,g^b=gi,h^b=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34838);