Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2396 conjugacy classes of subgroups; only the 538 normal subgroups are shown.
This group has 2396 conjugacy classes of subgroups — too many to draw. Only its 538 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,21)(23,24)(28,35)(32,37)(33,39)(38,40);(3,7)(6,8)(11,16)(13,15)(18,21)(22,23)(26,30)(31,36)(32,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(25,28,29,35)(26,32,30,37)(27,33,34,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=i,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371497);