Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25900 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62336 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,17,22,28)(5,23)(6,13)(7,8)(9,15)(10,20,21,30)(12,25)(14,18,26,32)(16,24)(19,27,29,31);(1,3,6,15,8,16,25,5)(2,9,12,11,7,23,13,24)(4,18,20,27,22,32,30,31)(10,26,28,19,21,14,17,29);(1,4,8,22)(2,10)(3,14,16,26)(5,19,15,29)(6,20,25,30)(7,21)(9,18)(11,27)(12,28)(13,17)(23,32)(24,31);(1,5,6,3,8,15,25,16)(2,11,12,23,7,24,13,9)(4,19,20,14,22,29,30,26)(10,27,28,32,21,31,17,18)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410700);