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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18,21,20)(19,22,24,23)(27,31)(29,32);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,13)(14,16)(17,18,21,20)(19,23,24,22)(25,26,28,30)(27,29,31,32);(1,3,4,8)(2,6,5,7)(9,12,14,15)(10,16,13,11)(17,19,21,24)(18,22,20,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20,21,18)(19,23,24,22)(25,27)(26,29)(28,31)(30,32);(1,3)(2,7)(4,8)(5,6)(9,13,14,10)(11,12,16,15)Polycyclic presentation:
2^9|a^2=fh,b^2=fhi,c^2=h,d^2=fh,e^2=g,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, 3288564);