Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #2543 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,15)(14,16)(17,18,24,21,23,27,22,32)(19,28,20,29,30,25,31,26);(3,8)(5,7)(9,11,13,10)(12,15,16,14)(17,19,23,30)(18,25,27,28)(20,22,31,24)(21,26,32,29);(1,3,4,8)(2,5,6,7)(10,11)(14,15)(17,20)(18,26)(19,22)(21,28)(23,31)(24,30)(25,32)(27,29);(2,6)(5,7)(10,11)(14,15)(17,21,24,27,23,32,22,18)(19,29,20,25,30,26,31,28);(9,12,13,16)(10,14,11,15)(17,22,23,24)(18,21,27,32)(19,20,30,31)(25,29,28,26)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6331677);