Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,18)(19,28)(20,22)(21,25)(23,26)(24,27)(29,30)(31,32);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,14,15,16)(17,19,24,31)(18,25,27,29)(20,28,26,32)(21,22,30,23);(1,4,5,2)(3,7,8,6)(10,15)(14,16)(17,20,23,27,24,26,22,18)(19,29,30,28,31,25,21,32);(3,8)(6,7)(9,12,13,11)(10,16,15,14)(17,21,22,31,24,30,23,19)(18,25,26,28,27,29,20,32)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ei,d^2=f,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,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343774);