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) = C25×C4 |
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 | |
2100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,13)(11,15)(17,18,21,20)(19,23,24,22)(25,26,29,31)(27,28,32,30);(1,3,6,5)(2,7,4,8)(11,15)(14,16)(25,26)(27,28)(29,31)(30,32);(1,2,6,4)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,24,23)(25,27,29,32)(26,30,31,28);(1,4,6,2)(3,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,20,23)(25,26)(27,30)(28,32)(29,31);(1,5,6,3)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,23,24,22)(25,28,29,30)(26,27,31,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=fh,c^2=ghi,d^2=fghi,e^2=ghi,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, 3764116);