Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2481 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)(9,10,13,15)(11,16,14,12)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,31,30,32);(5,6,7,8)(11,14)(12,16)(18,22)(19,26)(20,29)(31,32);(5,6)(7,8)(9,11,13,14)(10,12,15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31);(1,2)(3,4)(17,19,21,26)(18,23,22,25)(20,30,29,27)(24,32,28,31);(9,12)(10,14)(11,15)(13,16)(17,20,21,29)(18,24,22,28)(19,27,26,30)(23,31,25,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3942679);