Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434015 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15495 |
| 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 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,23,10)(3,13,15,8,5,22,24,26)(4,16,21,29,20,9,19,28)(14,30,17,25,18,27,31,32);(1,3,6,15,7,5,23,24)(2,8,10,13,12,26,11,22)(4,17,19,14,20,31,21,18)(9,27,28,32,16,30,29,25);(1,4,6,21,7,20,23,19)(2,9,10,29,12,16,11,28)(3,14,24,31,5,18,15,17)(8,25,22,27,26,32,13,30);(1,5)(2,8)(3,7)(4,18)(6,15)(9,25)(10,22)(11,13)(12,26)(14,20)(16,32)(17,21)(19,31)(23,24)(27,29)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 341211);