Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3726 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)(5,6,7,8)(9,10,13,16)(11,12,15,14)(18,23)(19,26)(20,25)(28,32);(9,11)(10,14)(12,16)(13,15)(17,18,22,23)(19,24,26,30)(20,21,25,29)(27,28,31,32);(5,6)(7,8)(9,12,13,14)(10,15,16,11)(17,19)(18,24)(20,27)(21,28)(22,26)(23,30)(25,31)(29,32);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,21)(19,27)(22,25)(23,29)(24,28)(26,31)(30,32);(9,10)(11,14)(12,15)(13,16)(17,21)(18,25)(19,28)(20,23)(22,29)(24,31)(26,32)(27,30)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^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, 4325968);