Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
3216 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)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,23,26,30)(20,24,29,28)(25,31,27,32);(5,6,7,8)(9,11)(10,14)(12,15)(13,16)(18,22)(23,30)(24,28)(31,32);(5,6)(7,8)(10,12)(14,15)(17,19,21,26)(18,23,22,30)(20,27,29,25)(24,32,28,31);(1,2)(3,4)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,32)(26,30)(27,31);(9,12,13,10)(11,15,16,14)(17,20,21,29)(18,24,22,28)(19,25,26,27)(23,31,30,32)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3242142);