Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18,22,26,21,24,30,29)(19,27,28,20,25,23,32,31);(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,29)(21,25)(22,28)(24,27)(26,31)(30,32);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(18,24)(19,25)(26,29)(28,32);(11,16)(13,15)(18,24)(19,25)(26,29)(28,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,20,30,27,21,31,22,23)(18,25,29,28,24,19,26,32)Polycyclic presentation:
2^9|a^2=h,c^2=fg,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,e^b=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314374);