Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501862 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24909 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33539 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1765 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
639 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)(3,14)(4,18)(5,21)(6,12)(7,13)(8,15)(9,17)(10,24)(11,19)(16,30)(20,29)(22,28)(23,27)(25,31)(26,32);(1,3,6,16,7,4,19,17)(2,8,11,27,13,29,12,28)(5,22,24,20,25,23,32,15)(9,21,18,26,30,31,14,10);(1,4)(2,9)(3,7)(5,20)(6,16)(8,26)(10,29)(11,14)(12,18)(13,30)(15,25)(17,19)(21,27)(22,24)(23,32)(28,31);(1,3)(2,8)(4,7)(5,23)(6,16)(9,26)(10,30)(11,27)(12,28)(13,29)(14,21)(15,24)(17,19)(18,31)(20,32)(22,25)Polycyclic presentation:
2^9|b^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,e^b=ehi,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409058);