Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,8,4,5)(9,10)(11,16)(12,18)(13,14)(15,17)(19,21)(20,23)(22,24);(1,3,6,4)(2,5,7,8)(9,11,15,14)(10,16,17,13)(12,20,22,19)(18,23,24,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(2,7)(5,8)(10,17)(11,14)(18,24)(19,20)(26,31)(29,39)(32,37)(33,40);(2,7)(3,4)(9,12,15,22)(10,18,17,24)(11,19,14,20)(13,23,16,21)(25,27)(26,32)(28,34)(29,33)(30,36)(31,37)(35,38)(39,40);(1,4,6,3)(2,8,7,5)(9,13)(10,11)(12,21)(14,17)(15,16)(18,20)(19,24)(22,23)(25,28)(26,33)(27,35)(29,37)(30,38)(31,40)(32,39)(34,36);(1,5,6,8)(2,3,7,4)(9,14,15,11)(10,13,17,16)(12,20,22,19)(18,23,24,21)(25,29,30,39)(26,34,31,35)(27,33,36,40)(28,37,38,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229190);