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 | |
4331 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)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,39)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,13)(10,15,16,18)(12,19,20,21)(17,22,23,24)(25,26,30,31)(27,28,32,35)(29,33,37,39)(34,36,38,40);(1,3,5,4)(2,7,6,8)(10,16)(15,18)(17,23)(22,24)(26,31)(28,35)(29,37)(34,38);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,27,30,32)(26,28,31,35)(29,38,37,34)(33,40,39,36);(1,3,5,4)(2,7,6,8)(9,13,14,11)(10,18,16,15)(12,19,20,21)(17,22,23,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,38)(34,39)(37,40);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,29,30,37)(26,33,31,39)(27,34,32,38)(28,36,35,40)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7982607);