Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16310 |
| 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 | |
860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(4,17)(5,6)(9,24)(10,26)(11,21)(14,29)(15,16)(18,30)(23,28)(25,31);(1,2,5,8,7,11,6,21)(3,13,12,9,16,22,15,24)(4,10,19,23,17,28,20,26)(14,27,29,25,30,32,18,31);(1,3,5,12,7,16,6,15)(2,9,8,22,11,24,21,13)(4,18,19,14,17,29,20,30)(10,27,23,25,28,32,26,31);(1,4,5,19,7,17,6,20)(2,10,8,23,11,28,21,26)(3,14,15,18,16,30,12,29)(9,25,13,27,24,31,22,32)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365323);