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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(3,7)(6,8)(9,11)(10,16)(12,14)(13,23)(15,20)(17,21)(18,22)(19,24)(26,27)(31,33)(32,34)(38,39);(1,2,5,4)(3,6,7,8)(11,20)(14,21)(16,24)(18,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,5,7)(2,6,4,8)(9,12)(10,13)(11,21)(14,20)(15,17)(16,18)(19,22)(23,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4,5,2)(3,8,7,6)(9,13)(10,17)(11,18)(12,19)(14,16)(15,22)(20,23)(21,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39);(9,14,15,21)(10,18,19,23)(11,12,20,17)(13,16,22,24)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325413);