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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10,12,14)(11,15,16,13)(19,23)(22,24)(26,28)(27,31);(1,3,5,8)(2,6,7,4)(9,10,12,14)(11,13,16,15)(27,31)(30,32);(2,7)(3,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,22,23,24)(25,26,29,28)(27,30,31,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31);(1,4)(2,8)(3,7)(5,6)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,24,23,22)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fhi,e^2=fhi,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, 3165181);