Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| 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 | #16508 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3062 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)(12,13)(14,16)(17,18)(19,24)(20,28)(21,30)(22,23)(25,26)(27,31)(29,32);(9,11,14,12)(10,15,16,13)(18,23)(21,28)(24,31)(26,32);(5,6,7,8)(10,16)(11,12)(17,19)(18,24)(20,29)(21,26)(22,27)(23,31)(25,30)(28,32);(1,2)(3,4)(5,6)(7,8)(9,12,14,11)(10,13,16,15)(17,20)(18,21)(19,25)(22,30)(23,28)(24,32)(26,31)(27,29);(9,13)(10,11)(12,16)(14,15)(17,21)(18,20)(19,26)(22,28)(23,30)(24,29)(25,31)(27,32)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4843409);