Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
2952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,20,27,23)(21,29,32,24)(26,31,30,28);(9,10)(11,13)(12,14)(15,16)(17,19,22,27)(18,23,25,20)(21,26,32,30)(24,31,29,28);(5,6,7,8)(9,10,12,14)(11,13,16,15)(17,19)(18,20)(21,30)(22,27)(23,25)(24,31)(26,32)(28,29);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,31)(22,23)(24,30)(25,27)(26,29)(28,32);(9,11,12,16)(10,13,14,15)(17,21,22,32)(18,24,25,29)(19,26,27,30)(20,28,23,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4932918);