Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,37)(39,40);(1,3,5,4)(2,6,8,7)(11,16)(13,15)(17,19)(18,22)(20,23)(21,24)(26,31)(27,33)(34,40)(35,37);(9,10,12,14)(11,13,16,15)(17,19,21,24)(18,23,20,22)(25,27,30,33)(26,32,31,28)(29,37,39,35)(34,36,40,38);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,22)(21,24)(25,28,30,32)(26,33,31,27)(29,38,39,36)(34,35,40,37);(1,4,5,3)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,22,24,23)(25,29,30,39)(26,34,31,40)(27,35,33,37)(28,36,32,38)Polycyclic presentation:
2^9|a^2=fhi,b^2=fhi,c^2=fg,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4201312);