Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1681 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)(3,7,8,5)(9,10,14,13)(11,16,12,15)(18,23)(19,20)(26,30)(27,33)(28,36)(38,40);(3,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,18,22,23)(19,21,20,24)(27,33)(28,36)(31,34)(32,37);(9,11,14,12)(10,15,13,16)(17,19,22,20)(18,24,23,21)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,11)(10,15,13,16)(17,20,22,19)(18,24,23,21)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,15,12,16)(17,21,22,24)(18,19,23,20)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6072072);