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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4290 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)(3,4)(5,8)(6,7)(9,10,15,19)(11,16,20,14)(12,17,21,24)(13,18,22,23)(25,26,30,32)(27,31,34,28)(29,33,37,39)(35,38,40,36);(1,3)(2,6)(4,8)(5,7)(9,11,15,20)(10,16,19,14)(12,22,21,13)(17,23,24,18)(25,27)(26,31)(28,32)(29,35)(30,34)(33,38)(36,39)(37,40);(1,2)(3,6)(4,7)(5,8)(11,20)(13,22)(14,16)(18,23)(26,32)(27,34)(29,37)(36,38);(1,4,5,6)(2,7,8,3)(9,12)(10,17)(11,13)(14,23)(15,21)(16,18)(19,24)(20,22)(25,28)(26,27)(29,38)(30,31)(32,34)(33,40)(35,39)(36,37);(1,4)(2,3)(5,6)(7,8)(9,13)(10,18)(11,21)(12,20)(14,17)(15,22)(16,24)(19,23)(25,29,30,37)(26,33,32,39)(27,35,34,40)(28,36,31,38);(1,3)(2,4)(5,7)(6,8)(9,14)(10,11)(12,23)(13,17)(15,16)(18,21)(19,20)(22,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948188);