Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2703 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2801 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)(11,15)(13,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32);(6,8)(9,10,12,14)(11,13,15,16)(18,22)(20,29)(23,25)(26,30);(5,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,29)(30,31);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,23,22,25)(20,30,29,26)(24,32,28,31);(9,11)(10,13)(12,15)(14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)Polycyclic presentation:
2^9|b^2=g,c^2=fh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2978325);