Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #2703 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1373 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)(17,18,24,27,23,21,22,32)(19,28,29,31,30,25,20,26);(3,8)(5,7)(9,11,13,10)(12,15,16,14)(17,19,23,30)(18,25,21,28)(20,24,29,22)(26,32,31,27);(1,3,4,8)(2,5,6,7)(10,11)(14,15)(17,20,23,29)(18,26,21,31)(19,22,30,24)(25,27,28,32);(2,6)(5,7)(10,11)(14,15)(17,21,24,32,23,18,22,27)(19,25,29,26,30,28,20,31);(9,12,13,16)(10,14,11,15)(17,22,23,24)(18,27,21,32)(19,29,30,20)(25,31,28,26)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,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, 6331669);