Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #2397 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1452 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)(11,16)(12,14)(13,15)(17,18,23,24,22,27,19,28)(20,21,30,32,31,25,26,29);(1,3,4,8)(2,5,6,7)(17,19,22,23)(18,24,27,28)(20,30,31,26)(21,29,25,32);(2,6)(5,7)(9,11,13,12)(10,14,15,16)(17,20)(18,25)(19,26)(21,27)(22,31)(23,30)(24,29)(28,32);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(20,31)(21,25)(26,30)(29,32);(2,6)(5,7)(17,21,23,32,22,25,19,29)(18,26,24,20,27,30,28,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314015);