Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509783 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31000 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,7,12,6,9)(3,13,17,26,19,15,16,8)(5,11,20,27,24,28,23,22)(14,21,30,29,31,32,25,18);(1,3)(2,8)(4,17)(5,21)(6,16)(7,19)(9,15)(10,13)(11,14)(12,26)(18,23)(20,29)(22,25)(24,32)(27,30)(28,31);(1,2)(3,14)(4,9)(5,22)(6,10)(7,12)(8,21)(11,23)(13,18)(15,29)(16,30)(17,25)(19,31)(20,28)(24,27)(26,32);(1,2)(3,15)(4,10)(5,11)(6,9)(7,12)(8,17)(13,19)(14,29)(16,26)(18,31)(20,27)(21,25)(22,23)(24,28)(30,32)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416979);