Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,24,19,23,28,22,27)(20,25,30,29,31,32,21,26);(1,2)(3,7)(4,8)(5,6)(17,19,24,28,23,27,22,18)(20,26,30,25,31,29,21,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,31)(18,25,28,32)(19,29,27,26)(21,24,30,22);(1,3,5,4)(2,8,6,7)(17,21,23,30)(18,26,28,29)(19,25,27,32)(20,22,31,24);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,27,28,19)(20,30,31,21)(25,29,32,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=df,d^c=di,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290931);