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 | |
4306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(25,26)(27,33)(28,30)(29,34)(31,32)(35,40)(36,37)(38,39);(1,2,5,6)(3,7,4,8)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(26,32)(27,28)(34,39)(35,36);(1,3,5,4)(2,7,6,8)(10,16)(15,18)(17,20)(23,24)(25,27,31,28)(26,33,32,30)(29,36,38,35)(34,37,39,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,28,31,27)(26,30,32,33)(29,36,38,35)(34,37,39,40);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,29,31,38)(26,34,32,39)(27,35,28,36)(30,37,33,40);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,30)(26,27)(28,32)(29,37)(31,33)(34,35)(36,39)(38,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621648);