Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #430117 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15413 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,21,7,9,6,11)(3,12,13,25,16,23,15,8)(4,17,19,27,20,10,22,28)(14,26,29,32,30,31,18,24);(1,3,5,13,7,16,6,15)(2,8,21,12,9,25,11,23)(4,18,19,14,20,29,22,30)(10,26,28,32,17,31,27,24);(2,9)(3,13)(5,6)(8,23)(10,17)(12,25)(14,29)(15,16)(18,30)(19,22)(24,31)(26,32);(1,4)(2,10,9,17)(3,14,16,30)(5,19)(6,22)(7,20)(8,24)(11,27,21,28)(12,26)(13,29,15,18)(23,31)(25,32)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337313);