Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2788 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)(9,10,13,15)(11,14,16,12)(17,18)(19,20)(21,29)(22,23)(24,25)(26,30)(27,32)(28,31);(6,8)(9,11,13,16)(10,14,15,12)(18,23)(19,25)(21,30)(28,32);(5,6)(7,8)(10,15)(12,14)(17,19)(18,24)(20,23)(21,31)(22,25)(26,28)(27,30)(29,32);(1,2)(3,4)(9,12)(10,11)(13,14)(15,16)(17,20,22,24)(18,25,23,19)(21,32,30,28)(26,27,29,31);(9,11)(10,12)(13,16)(14,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,32)(25,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4038486);