Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | #17082 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,24)(22,23)(26,29)(30,32);(10,11)(14,15)(17,19)(18,21)(20,22)(23,24)(27,31)(30,32);(12,16)(14,15)(17,20,23,21)(18,24,22,19)(25,26)(27,30)(28,29)(31,32);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,21,23,20)(18,24,22,19)(25,27)(26,30)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,22,23,18)(19,20,24,21)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|c^2=h,d^2=hi,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2481737);