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 | |
4656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,16)(11,17,20,15)(12,18,22,24)(13,19,23,21)(25,26,31,30)(27,34,29,32)(28,33,37,38)(35,40,36,39);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,19)(13,18)(14,20)(16,17)(21,22)(23,24)(25,27,31,29)(26,32,30,34)(28,36,37,35)(33,40,38,39);(10,16)(13,23)(15,17)(18,24)(27,29)(32,34)(35,36)(39,40);(9,11)(10,17)(12,21)(13,18)(14,20)(15,16)(19,22)(23,24)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,18,16,24)(11,19,20,21)(13,17,23,15)(25,29,31,27)(26,34,30,32)(28,35,37,36)(33,39,38,40);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,19,16,21)(11,18,20,24)(12,17,22,15)(25,30,31,26)(27,34,29,32)(28,33,37,38)(35,39,36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7988217);