Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,13)(12,15)(14,16)(17,18,21,22)(19,23,24,20)(27,31)(30,32);(1,3,6,4)(2,7,5,8)(10,11)(14,15)(17,19,21,24)(18,20,22,23)(25,26,29,28)(27,30,31,32);(9,10,13,11)(12,14,16,15)(17,18)(19,20)(21,22)(23,24)(25,26,29,28)(27,30,31,32);(1,4,6,3)(2,7,5,8)(9,11,13,10)(12,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,32)(29,31);(1,5,6,2)(3,8,4,7)(9,12,13,16)(10,14,11,15)(17,20,21,23)(18,19,22,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=fh,b^2=fghi,c^2=ghi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4835927);