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) = C25×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 | |
2319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(10,14)(11,16)(17,18)(19,21)(20,24)(22,23)(27,31)(30,32);(2,6)(3,8)(9,10)(11,15)(12,16)(13,14)(18,23)(19,20)(26,29)(30,32);(2,6)(4,7)(10,14)(12,15)(17,19,22,20)(18,24,23,21)(25,26)(27,30)(28,29)(31,32);(1,3,5,8)(2,7,6,4)(9,11,13,16)(10,15,14,12)(17,20,22,19)(18,24,23,21);(1,4,5,7)(2,8,6,3)(9,12,13,15)(10,16,14,11)(17,21,22,24)(18,19,23,20)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|c^2=fi,d^2=f,e^2=f,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, 4538488);