Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(11,15)(13,16)(17,18)(19,24)(20,29)(21,28)(22,23)(25,31)(26,32)(27,30);(9,10,12,14)(11,13,15,16)(18,23)(19,27)(25,29)(26,28);(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,27)(18,24,23,30)(20,28,31,26)(21,25,32,29);(1,2,3,4)(5,7,8,6)(17,20,22,31)(18,25,23,29)(19,28,27,26)(21,24,32,30);(9,11,12,15)(10,13,14,16)(17,21,22,32)(18,26,23,28)(19,25,27,29)(20,30,31,24)Polycyclic presentation:
2^9|b^2=fh,c^2=fg,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4342421);