Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C2×D42 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11)(4,6)(5,9)(7,10)(8,12)(13,15)(14,16)(17,18,22,25)(19,23,21,29)(20,27,28,31)(24,30,32,26);(1,3)(2,8)(4,11)(5,12)(6,14)(7,13)(9,15)(10,16)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(1,4)(2,5)(3,8)(6,10)(7,9)(11,14)(12,15)(13,16);(1,5,7,6)(2,9,10,4)(3,12,13,14)(8,15,16,11)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265606);