Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2522 conjugacy classes of subgroups; only the 546 normal subgroups are shown.
This group has 2522 conjugacy classes of subgroups — too many to draw. Only its 546 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)(27,33)(28,36)(30,34)(31,38);(3,7)(6,8)(11,16)(13,15)(18,21)(22,23)(25,26,29,32)(27,34,33,30)(28,31,36,38)(35,40,39,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27,29,33)(26,30,32,34)(28,35,36,39)(31,37,38,40);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,21)(22,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(25,28)(26,31)(27,35)(29,36)(30,37)(32,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=i,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, 4371453);