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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1866 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)(3,8)(4,6)(5,7)(10,12)(14,15)(18,22)(20,23)(25,26,28,30)(27,31,32,29);(1,3,5,6)(2,4,7,8)(9,10,13,12)(11,15,16,14)(18,22)(19,24)(27,32)(29,31);(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,29)(28,30)(31,32);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,19,21,24)(18,23,22,20)(25,27)(26,29)(28,32)(30,31);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=gi,e^2=h,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, 4802838);