Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10)(11,16)(12,13)(14,15)(18,22)(19,23)(26,27)(28,32);(9,11,14,13)(10,12,15,16)(17,18,21,22)(19,20,23,24)(26,27)(30,31);(10,15)(11,13)(18,22)(20,24)(25,26,29,27)(28,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19,21,23)(18,20,22,24)(25,27,29,26)(28,31,32,30);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,12,15,16)(17,20)(18,23)(19,22)(21,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|b^2=f,c^2=fh,d^2=fg,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4658666);