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 | #2456 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28,22,24,20,26)(19,29,30,32,27,25,31,21);(1,2,4,5)(3,7,8,6)(9,11,12,15)(10,13,14,16)(18,24)(21,32)(25,29)(26,28);(10,14)(13,16)(17,19)(18,25)(20,31)(21,28)(22,27)(23,30)(24,29)(26,32);(1,3,4,8)(2,6,5,7)(17,20,22,23)(18,26,24,28)(19,30,27,31)(21,29,32,25);(17,21,23,29,22,32,20,25)(18,27,28,31,24,19,26,30)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=dfi,e^b=ei,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, 6292820);