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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2539 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)(9,10,13,14)(11,12,15,16)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31);(5,6,7,8)(10,14)(12,16)(20,29)(23,28)(26,32)(30,31);(5,6)(7,8)(9,11,13,15)(10,12,14,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,2)(3,4)(9,12,13,16)(10,11,14,15)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32);(9,11)(10,12)(13,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fg,d^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, 3242134);