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) = C26 |
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 | |
1534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,13,11)(12,15,16,14)(17,18,21,20)(19,22,24,23)(25,26)(27,33)(28,36)(29,35)(30,31)(32,39)(34,40)(37,38);(9,10,13,11)(12,15,16,14)(17,19,21,24)(18,22,20,23)(26,31)(29,33)(32,36)(38,40);(10,11)(14,15)(17,19)(18,22)(20,23)(21,24)(25,27,30,35)(26,29,31,33)(28,37,39,34)(32,38,36,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(25,28,30,39)(26,32,31,36)(27,34,35,37)(29,40,33,38);(1,3,4,7)(2,5,6,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)(25,29)(26,27)(28,38)(30,33)(31,35)(32,37)(34,36)(39,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=f,d^2=g,e^2=fhi,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, 4186890);